Studies of Universality in Strongly Interacting Li Fermi ...\Studies of Universality in Strongly...
Transcript of Studies of Universality in Strongly Interacting Li Fermi ...\Studies of Universality in Strongly...
Studies of Universalityin
Strongly Interacting 6Li Fermi Gaseswith
Bragg Spectroscopy
A thesis submitted for the degree of
Doctor of Philosophy
by
Eva Dorothy Kuhnle
Centre for Atom Optics and Ultrafast Spectroscopy and
ARC Centre of Excellence for Quantum-Atom Optics
Faculty of Engineering and Industrial Sciences
Swinburne University of Technology
Melbourne, Australia
June 15, 2011
Declaration
I, Eva Dorothy Kuhnle, declare that this thesis entitled:
“Studies of Universality in Strongly Interacting 6Li Fermi Gases with
Bragg Spectroscopy”
is my own work and has not been submitted previously, in whole or in part, in
respect of any other academic award.
Eva Dorothy Kuhnle
Centre for Atom Optics and Ultrafast Spectroscopy
Faculty of Engineering and Industrial Sciences
Swinburne University of Technology
Melbourne, Australia
Dated this day, June 15, 2011
i
ii
Abstract
Universal relations for dilute, strongly interacting Fermi systems can be studied in
ultracold fermionic 6Li quantum gases using Bragg spectroscopy. Standard tech-
niques of laser trapping and cooling of atoms are applied to evaporate fermionic
gases below the critical temperature for superfluidity, TC . The atoms are prepared
to equally populate the lowest two spin states and the interaction between atoms
of opposite spin is precisely tunable via a Feshbach resonance. A unitary Fermi gas
represents a special case where the interactions are categorised as strong, that is,
the range of interaction exceeds the range of the interatomic potential. Elastic col-
lisions in the gas become unitarity limited and independent of details of the atomic
properties. This universality is a property of unitary Fermi gases and the univer-
sal relations adequately describe the thermodynamic behaviour of these gases. The
universal contact parameter is a central quantity that encapsulates the microscopic
details of the system and varies with the temperature and interaction strength.
The principal topic of this thesis is the use of Bragg spectroscopy as a tool to
measure the universal contact in trapped Fermi gases. The contact parameter quan-
tifies the short-range pair correlation function according to a universal law. Inelastic
Bragg scattering of photons allows one to measure the static structure factor, which
is by definition the Fourier transform of the pair correlation function. Due to this
well defined relation between the contact and Bragg spectra, the universal relation
for the static structure factor is experimentally verified for a range of transferred
momenta, k/kF = 3.5 − 9.1, in agreement with the theoretical predictions. The
contact in the zero-temperature limit at different interaction strengths as well as in
the unitarity limit at different temperatures agrees well with the calculations. The
contact at unitarity decreases monotonically for T/TF = 0 − 1 but has a non-zero
value well above the critical temperature TC ≈ 0.2 TF . The contact parameter may
offer new insights into criticality or complex superfluid pairing mechanisms such as
pseudogap-pairing.
iii
iv
Acknowledgements
The completion of this work involved big and small contributions from people whom
I met during my time at the Centre for Atom Optics and Ultrafast Technology
(CAOUS) in Melbourne. And therefore I would like to express my thanks to
Prof. Peter Hannaford for giving me the opportunity to work in the lithium lab
at CAOUS and to study strongly interacting Fermi gases, which were quite a terra
incognita to me when I started. I appreciate the opportunity to visit conferences
and the revision of this script.
Dr Chris Vale for the great supervision of my work and the freedom to ask all the
questions regarding theoretical and experimental matters. I very much value his
polite advice and patient support on basically everything. If things were ‘not quite
right’, he would ‘think about it’ and come up with ‘a good idea’.
Prof. Peter Drummond, Dr Hui Hu, Dr Xia-Ji Liu and Dr Chris Ticknor for pro-
viding these difficult calculations for the experiment and for being always patient
in answering my trivial questions. I was very fortunate to have had their support
and I learnt an enormous amount from them or was just impressed by their lateral
thinking.
Jurgen Fuchs, Gopi Veeravalli, Paul Dyke and Sascha Hoinka for their inspiring
attitude, practical implementations, commitment to the experiment and the bright
ideas. They were great colleagues and made it easy to work hard. I would also like to
thank Michael Mark and Michael Vanner for their contributions to the experiment
in making electronic and optical devices, calculations and good to-do lists.
The other members of CAOUS: Mark Kivinen for manufacturing the mechanical
parts, Tatiana Tchernova for organising everything and helping with the forms,
Wayne Rowlands and Tom Edwards for the opportunity to engage in teaching ac-
tivities, and all the PhD students at CAOUS whose company I very much enjoyed.
My parents for supporting and accompanying me in their thoughts all the way.
v
vi
Contents
Declaration i
Abstract iii
Acknowledgements v
Contents x
List of figures xii
Introduction 1
1 BEC-BCS crossover physics 7
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Low temperature scattering . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Scattering cross section and partial wave amplitude . . . . . . 10
1.2.2 Low energy s-wave scattering with interactions . . . . . . . . . 11
1.3 The broad Feshbach resonance in Lithium-6 . . . . . . . . . . . . . . 13
1.3.1 Scattering resonance . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Situation in lithium . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Thermodynamics of Fermi gases . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 Fermi distribution . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 The non-interacting homogeneous case . . . . . . . . . . . . . 18
1.4.3 The non-interacting trapped case . . . . . . . . . . . . . . . . 21
1.4.4 The interacting trapped case . . . . . . . . . . . . . . . . . . . 22
1.4.4.1 BEC limit . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.4.2 BCS limit . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.4.3 Unitarity regime . . . . . . . . . . . . . . . . . . . . 25
1.4.5 Hydrodynamic expansion of clouds . . . . . . . . . . . . . . . 25
vii
viii CONTENTS
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Universal relations 29
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Definitions of the contact . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 The contact and universal theorems . . . . . . . . . . . . . . . . . . . 32
2.3.1 Pair-correlation function . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Adiabatic sweep theorem . . . . . . . . . . . . . . . . . . . . . 36
2.3.3 Generalised virial theorem . . . . . . . . . . . . . . . . . . . . 37
2.4 Contact in the BEC-BCS crossover . . . . . . . . . . . . . . . . . . . 37
2.5 Universal contact at finite temperatures . . . . . . . . . . . . . . . . . 40
2.5.1 Thermodynamic contact from the entropy . . . . . . . . . . . 41
2.5.2 Contact for the trapped case at zero temperature . . . . . . . 41
2.5.3 Below the critical temperature . . . . . . . . . . . . . . . . . . 41
2.5.4 Above the critical temperature . . . . . . . . . . . . . . . . . 44
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Structure factor of a Fermi gas 47
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Bragg spectroscopy of pair-correlations . . . . . . . . . . . . . . . . . 48
3.2.1 Principle of Bragg scattering . . . . . . . . . . . . . . . . . . . 49
3.2.2 Advantages of Bragg spectroscopy . . . . . . . . . . . . . . . . 50
3.3 Dynamic structure factor . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Static structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5 f-sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6 Linear response function . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.7 Response at finite temperature . . . . . . . . . . . . . . . . . . . . . . 59
3.7.1 Consequences for the dynamic structure factor and the response 60
3.7.2 Detailed balancing . . . . . . . . . . . . . . . . . . . . . . . . 60
3.7.3 Fluctuation-dissipation theorem . . . . . . . . . . . . . . . . . 61
3.8 Dynamic structure factor from Bragg spectra . . . . . . . . . . . . . . 63
3.8.1 Measuring the dynamic structure factor . . . . . . . . . . . . . 64
3.8.2 Dynamic structure factor from the centre of mass displacement 64
3.8.3 Dynamic structure factor from the width . . . . . . . . . . . . 65
3.8.4 Application of sum-rules . . . . . . . . . . . . . . . . . . . . . 66
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Degenerate quantum gas machine 69
CONTENTS ix
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Feshbach coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Dipole traps for evaporation and experiments . . . . . . . . . . . . . 76
4.3.1 Principle of dipole traps . . . . . . . . . . . . . . . . . . . . . 77
4.3.2 Single focused dipole trap . . . . . . . . . . . . . . . . . . . . 77
4.3.3 Crossed dipole trap . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 Setup for Bragg spectroscopy . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Detection with absorption imaging . . . . . . . . . . . . . . . . . . . 83
4.5.1 Imaging hardware . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5.2 Image processing . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Temperature of unitary Fermi gases 91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Images for the temperature calibration . . . . . . . . . . . . . . . . . 93
5.3 Empirical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 Comparison with NSR theory . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Weakly interacting limit . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.6 Comparison of the methods . . . . . . . . . . . . . . . . . . . . . . . 102
5.7 Alternative approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Universal behaviour of pair correlations 107
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Universality of the static structure factor . . . . . . . . . . . . . . . . 108
6.3 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.2 Creating Bragg spectra . . . . . . . . . . . . . . . . . . . . . . 113
6.3.3 Dynamic response . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.4 Static response . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 Interaction dependence of the contact . . . . . . . . . . . . . . . . . . 119
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7 Temperature dependence of the contact 123
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3 Momentum and energy interrelation . . . . . . . . . . . . . . . . . . . 126
7.4 Bragg spectra after immediate release . . . . . . . . . . . . . . . . . . 128
x CONTENTS
7.4.1 Bragg spectra from the first moment . . . . . . . . . . . . . . 129
7.4.2 Bragg spectra from the second moment . . . . . . . . . . . . . 131
7.5 Bragg spectra after equilibration . . . . . . . . . . . . . . . . . . . . . 135
7.6 Temperature dependence of the contact . . . . . . . . . . . . . . . . . 139
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Conclusion and Outlook 143
Appendix 147
A Offset locking 147
Bibliography 170
List of publications 171
List of Figures
1.1 Temperature-coupling phase diagram . . . . . . . . . . . . . . . . . . 8
1.2 Tuning interatomic potentials . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Broad s-wave Feshbach resonance at 834 G . . . . . . . . . . . . . . . 16
2.1 Extracting the contact . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Energy versus momentum in Bragg scattering . . . . . . . . . . . . . 50
3.2 Dynamic structure factor for an ideal Fermi gas . . . . . . . . . . . . 53
3.3 Illustration of momentum transfer . . . . . . . . . . . . . . . . . . . . 54
4.1 Schematic setup of the apparatus . . . . . . . . . . . . . . . . . . . . 70
4.2 Laser locking frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Optical setup for cooling . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Optical setup for imaging . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Optical setup for trap laser . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6 Optical setup for Bragg spectroscopy . . . . . . . . . . . . . . . . . . 82
5.1 Image and profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Extracting cx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Temperature profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Temperature calibration . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Comparison with the weakly interacting regime . . . . . . . . . . . . 104
6.1 Bragg setup I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Tuning k/kF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 From images to line profiles to spectrum . . . . . . . . . . . . . . . . 114
6.4 Spectra from centre of mass displacement . . . . . . . . . . . . . . . . 116
6.5 Bragg signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.6 Verification of the universal static structure factor . . . . . . . . . . . 120
6.7 Interaction dependence of the contact . . . . . . . . . . . . . . . . . . 122
xi
xii LIST OF FIGURES
7.1 Bragg setup II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 Centre of mass displacement spectra after momentum transfer . . . . 130
7.3 Width spectra after momentum transfer . . . . . . . . . . . . . . . . 133
7.4 Dynamic structure factor after momentum transfer . . . . . . . . . . 134
7.5 Example of fitting a profile . . . . . . . . . . . . . . . . . . . . . . . . 136
7.6 Width spectra after energy transfer . . . . . . . . . . . . . . . . . . . 138
7.7 Dynamic structure factor after energy transfer . . . . . . . . . . . . . 138
7.8 Temperature dependence of the contact . . . . . . . . . . . . . . . . . 140
A.1 Schematic for offset locking . . . . . . . . . . . . . . . . . . . . . . . 149
Introduction
One of the most fundamental problems in physics is the many-body problem due
to its relevance to a diverse range of systems. Microscopic details are the origin
of bulk properties and the question is how these details influence the bulk proper-
ties. The many-body problem involves studies of the mutual interactions between
particles in a dense system and how they contrast with the behaviour of isolated,
noninteracting, and statistically distributed particles [1]. The resulting correlations
between interacting particles lead to collective modes whose significance is due to
the contribution of many particles.
Triggered by the discovery of superconducting mercury in 1911 [2], there has
been considerable interest in fermionic superfluidity and superconductivity, an effect
arising from many-body correlations. During the last 100 years, models of strongly
correlated systems have been developed that have shaped our modern understand-
ing of, for instance, the high-density electron gas found in solid-state devices [3, 4],
Bardeen-Cooper-Schrieffer (BCS) superconductivity [5], neutron stars and nuclear
structure (e.g. [6–8]), Bose-Einstein condensation [9–11], and dilute fermionic gases
with arbitrary interactions of finite range [12–16]. Dilute Fermi gases with strong
interactions are of broad interest, since their pairing behaviour forms a widely ap-
plicable paradigm.
The basic system considered here is a two-component Fermi gas of particles
with opposite spin, interacting only through a short-range potential. As long as the
gas is dilute and thermal, particles rarely collide and their motion is dictated by their
kinetic energy. The macroscopic behaviour is then classical and may be described
by classical gas laws found in standard textbooks. To reach quantum degeneracy
the temperature has to be reduced below a critical temperature where the Fermi-
Dirac statistics become pronounced [17]. The system would smoothly cross over
from a classical gas to an ideal quantum gas if there was no interaction present.
In practice, the cooling mechanism of the gas involves adiabatic rethermalisation
through collisional processes for the particles in the two different quantum states.
However, due to the Pauli exclusion principle the single state occupation number
1
2 INTRODUCTION
in a single-component Fermi gas is restricted to one particle per spin state and
the collisional cross section for particles with the same spin is strongly suppressed.
Higher order collisional processes are also frozen out due to the low temperatures;
thus only the lowest order collisional process for particles with different spin, the so-
called s-wave scattering, is relevant. The pursuit of fermionic quantum degeneracy
effects has led to the implementation of magnetic and optical traps that allow Fermi
gases to be cooled by evaporation of atoms in two different spin states [18], or as
two different isotopes [19,20] or as a mixture with Bose gases as refrigerants [21,22].
As a consequence of cooling, the kinetic energy diminishes and pairing effects
start to play a significant role; however, the superfluid phase transition occurs only
in the presence of interactions between spin-up and spin-down particles and the
interaction energy facilitates the pairing of fermions of opposite spin. Well known
examples of low temperature fermionic quantum superfluids are fermionic 3He [23],
neutron stars [8] and superconductors [5]. In ultracold atomic gases, the scattering
length, and hence the atomic interaction, can be tuned and enhanced with the help
of Feshbach resonances [24, 25]. Feshbach resonances are characterised by the two-
body interaction and permit one to change the magnitude as well as the sign of the
scattering length by simply varying an external magnetic field. Bosonic systems at
a large scattering length are accompanied by inelastic processes and subsequent loss
of atoms [26]; however, inelastic processes due to three-body losses are prohibited in
fermionic systems by the Pauli principle leading to a greater stability and lifetime
of the gas [27–29]. This advantage provides a testbed for investigating the variety
of pairing processes in the well-known crossover from fermionic superfluidity, where
the scattering length is small and negative [30,31], to the (molecular) Bose-Einstein
condensation of pairs, where the scattering length is small and positive [32–36]. The
scattering length diverges in the strongly interacting regime [37,38], often referred to
as the unitarity regime, where elastic collisions are quantum mechanically limited.
Experiments in the crossover region have been performed in which characteristic
quantities [35,39] and signatures of superfluidity such as collective properties [37,40,
41] and vortices [42] have been exploited. Since the first experimental investigations
in this regime in 2004 [31,35,36,41,43–45], progress has been extremely rapid with
studies of collective oscillations [40,46,47], universal behaviour [48–52], superfluidity
[42,53–56], polarised Fermi gases [52,57–61] and the speed of sound [62].
The present work focuses on the resonant regime at unitarity rather than the
whole crossover region. Strong interactions induce strong correlations of particles in
this many-body system. Many theoretical methods break down when dealing with
such strong correlations, as it is difficult to identify a small parameter appropriate
INTRODUCTION 3
for a perturbation analysis. However, the universality hypothesis [63] assumes that
the relevant length scale in the ground state at unitarity is the interparticle spacing
l = n−1/3, which is related to the density n and is much larger than the range of the
interatomic potential, l ≫ r0. Interactions at a short distance and with short-time
dynamics dominate over microscopic details. For example, at unitarity (a→ ∞) the
cross section reaches a constant value of 4π/k2, where k is the relative momentum of
the scattered atoms. Here, the many-body system becomes universal. This means
in turn that all dilute fermionic systems with sufficiently strong interactions behave
identically. For precisely this situation, exact universal relations have been derived
that accurately describe the behaviour of the system in any state: few-body or many-
body ground state, zero or nonzero temperature, homogeneous or inhomogeneous
gas, normal or superfluid states, balanced or imbalanced spin states. A number
of these relations were presented by Shina Tan [64–66] but efforts have since been
devoted to deriving these relations from other theoretical models or to deriving new
relations [67–74]. Tan showed that, in principle, the problem can be described by
modifying the many-body Schrodinger wave function for the case of an ideal gas with
an approximated contact interaction parameter. At the heart of these relations is a
quantity, the so-called contact, C, which is a measure of the number of closely spaced
fermions in two different spin states. The contact enters various properties such as
thermodynamic variables or large-momentum and high-frequency tails of correlation
functions. The contact is a key parameter which quantifies physical properties in the
phase diagram of the BEC-BCS crossover; therefore measurements of the coupling
and temperature dependence of the contact could be used to verify the Tan relations.
In this thesis, Tan’s pair-correlation function between spin-up (↑) and spin-
down (↓) atoms plays a central role. It diverges as C/r2 at short distances, r0 ≪r < 1/kF , where kF is the magnitude of the Fermi wave vector [64]. Correlation
functions are difficult to measure directly in ultracold gases; however, it is possible
to measure macroscopic quantities which depend on correlation functions in a well
defined way. By considering the pair-correlation function in Fourier space, one has
two advantages: (1) The macroscopic measurable quantity is the static structure
factor, S (k), which is simply given by the Fourier transform of the pair-correlation
function [75], and (2) pair-correlations in momentum space can be probed by Bragg
spectroscopy. Inelastic scattering is a well established technique to probe both the
dynamic and static structure factors of many-body quantum systems [76]. Ultra-
cold atoms are highly amenable to inelastic scattering through Bragg spectroscopy
which has previously been used to determine both the dynamic [26, 77] and static
structure factors of atomic Bose-Einstein condensates [78, 79]. In ultracold Fermi
4 INTRODUCTION
gases Bragg spectroscopy has been used by our group to determine both the dy-
namic and static structure factors over the BEC-BCS crossover, albeit at a single
momentum [80]. The inverse healing length determines the border between the high
momentum regime of single-particle excitations and the low momentum regime of
collective excitations, such as phonons or the Bogoliubov-Anderson mode [81]. Rel-
evant to the Tan relations is the high momentum limit k ≫ kF , where the pair
momentum distribution is inevitably linked to the contact n↑↓ (k) → C/k4 and the
static structure factor becomes [82]
S↑↓ (k ≫ kF ) =CnkF
kF4k
[1− 4
πkFa
kFk
]. (1)
The experimental verification of this exact and universal relation is the subject of this
thesis. It enables one to show the interaction dependence of the contact. In addition,
for the special case at unitarity, a→ ∞, the temperature dependence of the contact
is determined. The measurements are compared to new calculations of the contact
based on strong coupling theory [83–85]. The ability to vary the momentum over
a wide range differentiates Bragg spectroscopy from radio-frequency spectroscopy,
thus providing access to a broad range of the excitation spectrum and avoiding final
state interaction effects as no third atomic state is involved [56,86].
Thesis outline
Understanding unitary Fermi gases requires knowledge of the mechanisms causing
the strong interactions as well as the associated thermodynamics. Therefore, start-
ing with some theoretical background, chapter 1 recalls the basic scattering physics
required to understand the role of strong interactions in ultracold Fermi gases. The
BEC-BCS crossover provides a testbed to study pairing mechanisms. The thermo-
dynamics in this regime are dominated by the statistical nature of Fermi gases and
the characteristic Fermi-Dirac distribution determines the length and energy scales
of the system.
Chapter 2 is devoted to the universal relations. Starting with an emphasis
on universality in Fermi gases, the contact parameter and the Tan relations are
introduced as pioneering work. Their ability to explain the thermodynamic relations
is a powerful property, and in this chapter the temperature dependent contact at
unitarity is reviewed.
The experimental tool of choice to measure short-range correlations is Bragg
spectroscopy. In order to provide motivation for the use of this method an intro-
duction on linear response theory is given in chapter 3. The dynamic and static
INTRODUCTION 5
structure factors represent the measurable quantities provided by Bragg spectra, and
can be extracted from the centre of mass displacement as well as from the width
of the atomic distribution. In order to determine the absolute value of the static
structure factor from Bragg spectra, sum rules are necessary.
The experimental apparatus is described in chapter 4. Most importantly,
in order to conduct the following experiments precise control is required over the
Fermi momentum and hence the trap frequencies. Imaging is carefully adjusted
as obtaining good signal-to-noise is crucial in the Bragg spectra as well as for the
determination of the temperature of the Fermi gases.
Chapter 5 describes the temperature determination of Fermi gases at unitar-
ity. This topic is still a major challenge in current research requiring careful discus-
sion, and a full chapter is devoted to this. The determination of the temperature is
relevant for the results in the final experimental chapter.
The verification of the universal law for the static structure factor, S (k), versus
momentum transfer and its experimental procedure are the subject of chapter 6.
The construction of Bragg spectra and their analysis are described. Having de-
veloped a well defined procedure to determine the absolute static structure factor,
the universal relation is used to show the coupling dependence of the contact for a
zero-temperature approximation.
The final chapter 7 combines previously addressed theoretical knowledge and
experimental techniques to measure the temperature dependence of the contact at
unitarity. Bragg spectra of strongly interacting Fermi gases are taken at increasing
temperatures. Two comparative methods are used to obtain these spectra and
the resulting values of the contact obtained from the static structure factors are
compared with the theoretical predictions.
6 INTRODUCTION
Chapter 1
BEC-BCS crossover physics
Ultracold strongly interacting Fermi gases provide a paradigm to investigate phe-
nomena such as superfluidity and universality. In the crossover region, pairing mech-
anisms change from fully spatially correlated to fully momentum correlated, which
is represented in the temperature-coupling phase diagram where Bose-Einstein con-
densates (BEC) change to Bardeen-Cooper-Schrieffer superfluids (BCS). In the uni-
tarity regime, both the spatial and momentum correlations are comparable. This
characteristic length scale of the system is smaller than the zero-energy scattering
length and larger than the potential range of the interparticle interaction. Then,
this principle of a dilute and strongly interacting Fermi gas is a model applicable to
a broad range of physical systems like neutron stars or solid state physics.
1.1 Introduction
An important goal in ultracold quantum gases is to find novel macroscopic quantum
states to investigate superfluidity and universality. In the following we consider a
gas of fermions with an equal number of spin-up and spin-down particles. The two-
body problem involves interaction between two particles of opposite spin. The degree
of correlation of these pairs will then influence the interparticle collisions and will
therefore characterise the macroscopic properties of the gas. The relevant quantities
that modify these correlations are the temperature, T/TF , in terms of the Fermi
temperature, TF , and interaction strength, 1/(kFa). The latter is a quantity given
by the Fermi wave vector, kF , and the so-called s-wave scattering length, a. In the
scattering problem at low temperatures only the lowest partial wave survives and a
becomes the quantity of interest to describe two-body interactions. The interaction
strength 1/(kFa) is sometimes referred to as the coupling constant. Depending on
the temperature and scattering length, the macroscopic character of the gas changes
7
8 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
Figure 1.1: Sketch of the phase diagram as a function of the temperatureand interaction strength. Pairs of composite fermions change their correlationrange throughout the Bose-Einstein condensate (BEC) to Bardeen-Cooper-Schrieffer(BCS) crossover non-trivially from strong (light copper) to weak (dark copper). Thedashed line indicates the onset of pairing, T ∗. The solid line represents the criticaltemperature, TC .
drastically. Figure 1.1 illustrates the temperature-coupling phase diagram. In the
limit of a small and positive scattering length, a→ +0, tightly bound pairs are highly
spatially correlated. They attain a bosonic character and can undergo Bose-Einstein
condensation below a critical temperature. For a small and negative scattering
length, a → −0, the gas becomes a Fermi liquid described by the Bardeen-Cooper-
Schrieffer theory for superfluids. Momentum correlated atoms in the two spin states
are then also known as Cooper pairs. The mechanism for superfluidity evolves
smoothly from the BEC side to the BCS side. At zero temperature, the ground
state of a balanced gas is always superfluid. In the region where the scattering
length diverges the interactions are the strongest and quantum-mechanically limited.
This unitarity regime obeys universal relations and aroused considerable interest in
atomic physics. The critical temperature, TC , for fermionic superfluidity in this
limit is about 0.2 T/TF , and is many orders of magnitude higher than in the non-
interacting limits, a → ± 0. Also, the critical velocity, vC , is much higher. Both
effects allow superfluidity to be investigated away from the ground state.
This chapter reviews the influence of the interaction strength and temperature
on the pairing mechanisms in Fermi gases throughout the crossover regime from the
1.2 LOW TEMPERATURE SCATTERING 9
limiting cases of spatially correlated pairs to momentum correlated pairs. Since the
temperature and coupling strength form a two-fold parameter space, the structure
of this chapter starts with the introduction of the scattering length, a, in section 1.2,
which dominates the collisional properties of degenerate quantum gases at ultra-low
temperatures. The maximum of the s-wave scattering amplitude and a change of the
sign of the phase occurs at the Feshbach resonance, which is explained in section 1.3.
Therefore, the unitarity regime can be realised with Feshbach resonances. These
Feshbach resonances are a physical effect that allow control over the magnitude and
sign of 1/(kFa) in terms of the scattering length by varying the external magnetic
field. The isotope 6Li, which is used in this work, is a promising candidate [87,
88] due to its large and negative background scattering length. 6Li provides an
exceptionally broad Feshbach resonance; hence the magnetic field can be used to
precisely manipulate ultracold collisions and to investigate the whole BEC-BCS
crossover. Once the theoretical background for the unitarity regime is built, the
thermodynamic aspects for this regime are summarised in section 1.4.
In unitary Fermi gases, a clear definition of the length scales is important.
For the remainder of this thesis, the following definitions apply for the scale of the
characteristic length, r, and momentum, k, where r ∝ 1/k:
• The Fermi gas is ultracold and r must be smaller than the thermal deBroglie
wavelength, λdB =√~2/(2πmkBT ), with mass m, Boltzmann constant kB at
a temperature T .
• The s-wave scattering length at the broad Feshbach resonance, |a0| ≡ a, is
larger than r, see section 1.3.
• The interparticle distance, l, is larger than r.
• The inter-fermionic potential range, r0, is smaller than r.
Therefore, the restrictions on the length scales are r0 ≪ k−1 ≪ l ≪ λdB ≪ a.
1.2 Low temperature scattering
The interactions in a many-body system are determined by the scattering behaviour.
Scattered particles not only give information about the inner structure of materials,
see for instance chapter 3, but also are responsible for establishing the equilibrium
state of the system through elastic and inelastic collisions; for example, rethermalisa-
tion during the evaporative cooling process is a collisional process. These collisions
10 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
have a different influence on the dynamics of the gas depending on the involved
particles (bosons or fermions). Here, only the two-body interactions are discussed
because in a many-body system they provide the essential properties and any fur-
ther change can be covered by a perturbation from the mean field of the many-body
background. In this section, the relevant quantities in scattering events are recalled.
Detailed discussions can be found in textbooks, e.g. [89].
1.2.1 Scattering cross section and partial wave amplitude
The scattering process will hitherto be described by the potential V (x) and is located
in the centre of the coordinate system. Two-body scattering problems are commonly
described in the centre of mass frame as only the relative motion is relevant reducing
the degrees of freedom from six to three. Let us consider the collision between two
identical particles. At t0, the incoming wave is a plane wave, or a particle with
momentum, ~k, and reduced mass, m. The outgoing wave function consists of
an unscattered part and a scattered part. The exact eigenstates, ψk(x), after the
scattering event have to obey the Schrodinger equation[− ~2m
∇2 + V (x)
]ψk(x) = Ekψk(x), (1.1)
with the potential V (x) and kinetic energy
Ek =~2k2
2m≥ 0. (1.2)
Solutions with E < 0 support discrete bound states with a binding energy, EB. The
formal solution of the general structure of the stationary states ψk(x) leads after
some calculation to
ψk (x) = ψk,in (x) + ψk,sc (x) = eik·x +eikr
rfk (θ, ϕ) . (1.3)
The scattering amplitude, fk, depends on the scattering angle θ but not on the
distance and has the dimension of length. The total cross section is the integration
over all solid angles Ω
σ =
∫dΩ |fk(θ, ϕ)|2 . (1.4)
The scattering amplitude can be simplified by expanding the plane wave in
partial waves and using symmetry arguments, i.e. considering the spherical symme-
try of the problem and hence the independence of ϕ. Then, the scattering amplitude
1.2 LOW TEMPERATURE SCATTERING 11
may be written as
fk(θ) ≡ f(k, θ) =∞∑l=0
(2l + 1)fl (k)Pl(cos (θ)). (1.5)
The expansion coefficients fl are called the partial wave amplitudes which are given
by the phase shift δl (k) for the lth partial wave
fl (k) = eiδl(k)sin (δl (k))
k. (1.6)
Using the optical theorem, the scattering amplitude can be written as
fl(k) =1
k cot (δl (k))− ik. (1.7)
Expansion in terms of the effective range, rl, and the scattering length, al, leads to
k cot (δl (k)) = − 1
al+rl2k2 +O
(k4). (1.8)
Each partial wave follows from equations 1.4, 1.5 and 1.6 as
σl (k) =4π
k2(2l + 1) sin2 (δl (k)) . (1.9)
Since the cross section is additive, the total cross section can be calculated from
σ =∑
l σl.
1.2.2 Low energy s-wave scattering with interactions
The above scattering expressions can be modified for ultracold gases. The kinetic
energy is low, thus k → 0. The phase shift approximates δl (k) ∝ k2l+1 modulo
π and all partial waves with l = 0 have a vanishing cross section according to
σl (k) ∝ k4l. Phenomenologically, this is due to a reflection of the free wave function
from an additional centrifugal barrier in the effective scattering potential. For l = 0,
the phase shift must be δ0 (k) = −ka0 and the cross section is dominated by pure,
isotropic s-wave scattering. For the remainder of this thesis, the s-wave scattering
length is denoted as a0 ≡ a.
In the Schrodinger equation 1.1 of a many-particle system, an effective poten-
tial for the interactions can be expressed by a zero-range contact potential introduced
12 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
by Lee, Yang and Huang [90]
Veff (r) =2πa~2
mδ (r)
∂
∂rr. (1.10)
This condition is sometimes contrasted with the Bethe-Peierls boundary condition
[91] [1
rψ
d (rψ)
dr
]r=0
= −1
a. (1.11)
Both equations constrain the wave function properly in the zero-range limit. This
aspect is important when two fermions are close enough to each other to interact or
form a dimer, depending on the sign of the scattering length. Under these assump-
tions, the right hand side of equation 1.8 reduces to −1/a0 = −1/a in the zero range
limit and equation 1.7 becomes
f0(k) = − a
1 + ika. (1.12)
Poles occur for the condition −1/al = ik, in general. The real part of −f0(k) is
sometimes denoted as the effective scattering length, aeff = a/(1 + k2a2). The
scattering length follows as
a = − limk→0
tan (δ0 (k))
k. (1.13)
The total cross section for two identical particles is obtained from
σ =4πa2
1 + k2a2. (1.14)
For the weakly interacting limit, ka≪ 1, the cross section reads
σ (k → 0) = 4πa2, (1.15)
which is four times the classical scattering cross section and equal to the surface
area of the sphere, i.e. scattering occurs from the entire surface. Equation 1.15 also
shows that the low-energy scattering process does not depend on the details of the
microscopic potential. In the strongly interacting limit, ka≫ 1, the scattering cross
section becomes independent of the scattering length giving the maximum possible
cross section for s-wave collisions
σ (a→ ∞) =4π
k2. (1.16)
1.3 THE BROAD FESHBACH RESONANCE IN LITHIUM-6 13
This is the unitarity regime which can be realised in a Fermi gas at the BEC-BCS
crossover.
1.3 The broad Feshbach resonance in Lithium-6
The broad s-wave Feshbach resonance in 6Li at 834 G forms a rich testbed for the
investigation of correlated fermions [92]. Broad s-wave Feshbach resonances have
been subject to several experiments using either 6Li [31, 35, 36, 44, 45] or 40K [43].
These were initially studied in nuclear scattering mechanisms. The terms ‘open’,
‘closed channel’ were introduced and used to describe the picture of an effective
potential [93, 94] as well as the resonances between discrete states coupled to a
manifold of scattering states [95]. Originally proposed for hydrogen [96], the idea
was extended to ultracold alkali atoms [97]. Early interest concerned the modi-
fication of elastic and inelastic atomic collisions [24, 25, 98]. The observed strong
loss mechanisms in Bose gases [25, 26] were explained with the creation of large
ultracold diatomic molecules [99–101] having a size of the order 1000 a0. For
fermionic gases, the collisional relaxation into deeply bound states is suppressed
by the Pauli exclusion principle despite their highly excited vibrational state [27].
Consequently, a fermionic gas is stable close to a Feshbach resonance [102–105].
Important experiments to understand pairing mechanisms have been performed in
the vicinity of a Feshbach resonance, for instance the formation of stable diatomic
molecules [18, 28–30, 37, 38, 43, 106, 107] and the condensation of molecules [32–34].
The first production of cold dimers was demonstrated by photo-associating cold
atoms [108]. After considering the possibility to create ultracold molecules near
Feshbach resonances [100], the first signature of molecules created near a Feshbach
resonance were seen through Ramsey fringes between molecules and atoms [102]
and directly in 40K after adiabatically sweeping across a Feshbach resonance [30].
Molecule creation with 6Li followed in the same year [18, 28, 29]. Nowadays, it has
become common to cool and evaporate close to the Feshbach resonance to generate
a molecular BEC through weakly bound diatomic molecules of two fermions. More
details can be found in review articles about Feshbach resonances, e.g. [109].
1.3.1 Scattering resonance
In this context, the Feshbach resonance is understood as a scattering resonance. We
consider two free atoms in an arbitrary state. Their interatomic potential curve is
shown in figure 1.2. The open channel refers to the interatomic potential of the two
14 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
Figure 1.2: The Feshbach resonance results from tuning the potential curve fortwo interacting atoms in the open channel into resonance with the potential curvein the closed channel.
free atoms at large atomic distances (R = ∞). If the two states (molecule and two
free atoms) have a different magnetic (electric) dipole moment, µmolecule = µ2atoms,
an external magnetic (electric) field, B, can be used to detune the interatomic
potential, ∆E ∝ ∆µB. Then, the closed channel represents the energy-detuned
potential such that the highest lying bound state is in resonance with the energy of
the two free atoms. That is the threshold energy of the colliding atoms in an open
channel becomes equal to a bound energy state in the closed channel. The molecular
state exists for a limited lifetime. The scattering length of such a resonance depends
on the applied external magnetic field, B, and is generally given by
a = abga0
(1 +
∆B
B −B0
)(1 + α(B −B0)) (1.17)
with the background scattering length of the open channel abg in units of the Bohr
radius a0 = 5.292 10−11, the width of the resonance ∆B, which depends on the
magnetic moments of the closed and open channels, and the resonant magnetic field
B0 as well as the correction factor α.
1.3 THE BROAD FESHBACH RESONANCE IN LITHIUM-6 15
1.3.2 Situation in lithium
Lithium is an alkali element with a single valence electron in its ground state (S =
1/2 and L = 0, i.e. J = S +L = 1/2). The fermionic isotope 6Li has a nuclear spin
of I = 1, leading to a total angular momentum F = I ± J , which is F = 1/2 and
F = 3/2. The hyperfine states split at high magnetic fields (Paschen-Back regime)
depending on the projection quantum numbers mF . The broad resonance occurs for
an interatomic potential between the substates |F,m⟩ = |1/2, 1/2⟩ and |1/2,−1/2⟩(triplet state S = 1). The exact resonance position is B0 = 834.15 G (predicted
by [110]) with a width of ∆B ≈ 300G [39]. Another narrow resonance is found at 543
G and is ≈ 100 mG wide (singlet state S = 0) [106]. Here, kF |r0| ≥ 1, the effective
range of interactions, r0, is dominating [92]. The background scattering length for
the broad resonance is abg ≈ −1405a0 and abg ≈ − 2100 a0 (triplet scattering
length) at high fields. This is very unusual compared to other alkali elements where
abg ≤ 100a0. The large background scattering length modifies the continuum states
such that two colliding atoms are likely to be found close together and leads to
a large Franck-Condon factor (good overlap of the free continuum state and the
closed channel bound state). In this experiment, a magnetic field is generated by a
pair of Helmholtz coils and is used to precisely control the scattering length across
the Feshbach resonance according to the relation equation 1.17. Figure.1.3 sketches
the scattering length versus magnetic field. Although not shown, the scattering
length starts off near zero in zero magnetic field. With increasing magnetic field the
scattering length decreases to a local minimum of − 300 a0 at 325 G before crossing
zero at 530 G. The correction factor in equation 1.17 is α = 0.0004 for 6Li.
1.3.3 Bound states
Feshbach bound states for a three-dimensional Fermi gas exist only for repulsive
interactions, ka > 0. As discussed above, if the magnetic moment of a close channel
is different from the open channel, these can be tuned into resonance by the magnetic
field, where the resonance interaction is given by the scattering length a [30]. As
found by Petrov, the characteristic size of a Feshbach dimer is the scattering length
a [27]. The relaxation into a deeply bound molecular state requires three fermions
within a distance r0 to collide. As two of these three fermions must have the same
spin, the relative wave function has to be antisymmetric; thus in the node where
r = 0, the wave functions scales as kr, where r is small and the relative momentum
k ∝ 1/a goes with the inverse of the scattering length. Within the conditions of a
zero-range model, see equations 1.10 and 1.11, and a positive scattering length, the
16 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
500 600 700 800 900 1000 1100−10
−5
0
5
10
Magnetic field
Sca
tterin
g le
ngth
(10
00 a 0)
Figure 1.3: Broad s-wave Feshbach resonance at 834 G in 6Li. The scatteringlength diverges over a broad range and can therefore be precisely tuned with themagnetic field. This opens a way to investigate the BEC-BCS crossover.
wave function of the bound state is
Ψb =e−r/a
√2πar
. (1.18)
The binding energy, Eb, of the dimer does not depend on the short-range details of
the potential
Eb = − ~2
2ma2, (1.19)
where m is the mass in the centre of mass framework. The smaller the binding
energy, the larger the dimer. Several collisional processes between a combination of
dimers and atoms can take place once the molecular state is populated. On the one
hand, when the scattering length approaches resonance, 1/a→ 0, the binding energy
vanishes. Due to the Pauli principle, elastic collisions dominate. On the other, for
very deep binding energies, ka → +∞, and a molecular state far below the atomic
continuum, a loss channel opens for atom-dimer collisions. It has been shown that
the scattering length for dimer-dimer collisions add = 0.6 a and dimer-atom collisions
aad = 1.2 a can be related to the atom-atom scattering length a [27, 111]. But the
rate constant of inelastic atom-dimer collisions is αad ∝ (r0/a)3.33, while for dimer-
dimer collisions it is αdd ∝ (r0/a)2.55. Nevertheless, as r0/a → 0, the dimer-dimer
collision rate exceeds the atom-dimer collisions, allowing efficient cooling of dimers
1.4 THERMODYNAMICS OF FERMI GASES 17
close to resonance.
1.4 Thermodynamics of Fermi gases
The next sections discuss the basic thermodynamic quantities required to under-
stand the physics occurring in degenerate Fermi gases. Many textbooks (e.g. [112])
provide detailed discussions, derivations, and relations for Fermi gases; to quote
these is beyond the scope of this thesis and only a summary is given. As opposed
to textbooks, the details on the Fermi-Dirac statistics are introduced only now,
because the dependence on the temperature introduces another parameter in the
problem. A spin balanced Fermi gas is considered unless stated differently. A set of
equivalent, fundamental relations between energy, temperature and interaction shall
describe the phase diagram and is given by E/NϵF (T/TF , 1/(kFa)), see figure 1.1.
Energy and temperature are obviously macroscopic quantities. Though the interac-
tion strength is a two-body quantity, in the many-body problem one can treat the
two-body effects as central mechanisms which are weakly perturbed by the mean
field. The Fermi-Dirac distribution enters thermodynamic equations including their
derivations, see section 1.4.1, and the Fermi energy sets the relevant energy scale.
First, the ideal gas case is considered in the homogeneous case and the trapped case,
in section 1.4.2 and 1.4.3, respectively, since this marks a reference point in contrast
with the interacting cases, in section 1.4.4. Then the ideal (non-interacting) trapped
case is reviewed. The ideal case can often be used to describe the weakly interacting
regime (BCS side) of the crossover. The thermodynamic quantities in the ideal gas
case are valid for the equilibrium state and collisions are responsible for rethermal-
isation. The Fermi gas at zero temperature is not superfluid until interactions are
included. It is peculiar that the unitarity regime shows the same behaviour as the
ideal case within a rescaling. The chapter ends with a discussion of hydrodynamic
expansion as shown by time-of-flight experiments, section 1.4.5.
1.4.1 Fermi distribution
Thermodynamics studies the macroscopic behaviour of gases in the thermodynamic
limit, i.e. N ≫ 1. At large N the fluctuations ∆N are small and a statistical de-
scription becomes increasingly exact. The replacement of the temporal progression
of the microscopic details in the gas with the parallel existence of all possible states
is formulated in the ergodicity hypothesis; thus thermodynamics becomes thermo-
statistics. Extensive and intensive thermodynamic potentials such as entropy, tem-
18 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
perature, internal energy etc. describe the state of the gas macroscopically and their
first and second derivatives are linked (Maxwell relations) such that the potentials
can be re-constructed from each other. The information about the macroscopic state
is stored in the grand partition function in either the micro-canonical, canonical or
grand-canonical case, i.e. depending on a connection to an outer thermodynamic
environment which would allow temperature and/or particle exchange. In the grand-
canonical case of Fermi gases the grand partition function Z is a product state that
obeys asymmetry rules and the Pauli principle
ZFD = (1 + f1) (1 + f2) ... (1 + fk) ... =∞∏i=1
(1 + fi) (1.20)
with fi = e(µ−ϵi)/(kBT ). The grand potential can be obtained from
ΩFD = −kBT lnZFD = −kBT∞∑i=1
ln(1 + e(µ−ϵi)/(kBT )
)(1.21)
and the occupation number of the energy levels is
fFDk =
∂ΩFD
∂ϵk=
1
e(ϵk−µ)/(kBT ) + 1(1.22)
known as the famous Fermi-Dirac distribution. This is the signature to all mecha-
nisms in Fermi gases.
1.4.2 The non-interacting homogeneous case
In an ideal Fermi gas the number of states can be obtained by counting the states
in an octant of the momentum sphere with radius k (ϵ) =√
2mϵ/~2, spin degree of
freedom 2, and volume 4π/3 (π/L)3 (length L). Deriving this with respect to the
energy limit ϵ gives the density of states
D (ϵ) =2V
4π2
(2m
~2
)3/2√ϵ. (1.23)
When integrating over all energy states the Fermi distribution in equation 1.22 be-
comes
f (ϵ, T, µ) =1
e(ϵ−µ)/(kBT ) + 1. (1.24)
1.4 THERMODYNAMICS OF FERMI GASES 19
Using equation 1.21 the grand potential is
Ω = −kBT∫ ∞
0
ln(1 + e(µ−ϵi)/(kBT )
)D (ϵ) (1.25)
= −kBT2V
4π2
(2m
~2
)3/2 ∫ ∞
0
ln(1 + e(µ−ϵi)/(kBT )
)√ϵ,
which is unfortunately not analytically solvable. The atom number arises from
N = −(∂Ω
∂µ
)T,V
=
∫ ∞
0
f (ϵ, T, µ)D (ϵ) dϵ (1.26)
N (T, V, µ) =2V
4π2
(2m
~2
)3/2 ∫ ∞
0
√ϵ
1 + e(µ−ϵ)/(kBT )dϵ.
The internal energy is
U =
∫ ∞
0
ϵf (ϵ, T, µ)D (ϵ) dϵ (1.27)
U =2V
4π2
(2m
~2
)3/2 ∫ ∞
0
ϵ3/2
1 + e(µ−ϵ)/(kBT )dϵ.
A partial derivation shows Ω = −23U = −PV and therefore the equation of state of
a Fermi gas is
P =2
3
U
V(1.28)
For high temperatures or low particle densities, the fugacity is exp (µ/(kBT )) ≪ 1
and equation 1.24 reduces to the classical Boltzmann statistics as the exponential
term in the denominator exceeds the contribution +1:
f (ϵ, T, µ)e(µ/(kBT))≪1→ e(ϵ−µ)/(kBT ) (1.29)
If the term ‘exp(µ/(kBT ))’ is of the order of 1, quantum effects come into play. For
T ≤ 1 mK the deBroglie wavelength λdB =√
2π~2/(mkBT ) becomes comparable
to the interparticle spacing, where m is the mass of the particle at temperature
T , and the phase space density D = nλ3dB depends on the particle density. When
the temperature approaches T = 0 the Fermi distribution becomes more and more
similar to a step function, i.e. the occupation of available phase space cells smoothly
20 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
approaches unity
f (ϵ, T, µ) =1
e(ϵ−µ)/(kBT ) + 1→
1 for ϵ ≤ µ and T → 0,
0 for ϵ ≥ µ and T → 0
. (1.30)
The chemical potential is then a temperature dependent quantity. The particle
number is
N =2V
4π2
(2m
~2
)3/2 ∫ µ0
0
√ϵdϵ =
V
3π2
(2m
~2
)3/2
µ3/20 . (1.31)
The chemical potential is for a given particle number density-dependent
µ0 ≡ ϵF ≡ kBTF =~2
2m
(3π2n
)2/3 ∝ n2/3 (1.32)
So, at zero temperature, µ is the energy of the highest occupied state of the non-
interacting Fermi gas and equals the Fermi energy ϵF . This is referred to as the
ground state of the system where the lowest single particle states are filled up gap-
less and successively up to the Fermi energy ϵF . Any excitations into a higher lying
state makes the step function smoother by creating a ‘particle-hole’ relation. Con-
sequently, the width of the step increases at higher temperatures. The width is set
by the temperature, ∆ϵF = 2kBT . The ground state energy of the system is
E (T = 0) =
∫ µ0
0
ϵD (ϵ) dϵ =3
5Nµ0 =
3
5NϵF (1.33)
Therefore, in the homogeneous case the mean energy per particle at T = 0 is
E
N=
3
5ϵF (1.34)
which is different from the classical gas case, where E = 3kBT/2 and would be zero
at T = 0. However, the classical gas case is not valid at arbitrary low temperatures.
In Fermi systems, thermodynamic quantities depend weakly on the temperature,
because of the Pauli principle. The low lying states∣∣σ±1/2,k
⟩with small wave
vectors are only occupied once and many particles have to sit in higher energetic
states. For temperatures 0 ≤ T ≪ TF , the step function becomes continuously a
smoother transition between 0 and 1.
1.4 THERMODYNAMICS OF FERMI GASES 21
1.4.3 The non-interacting trapped case
Let us consider an ultracold Fermi gas of N↑ = N/2 spin polarised fermions in a
harmonic trapping potential
V =1
2mω2
xx2 +
1
2mω2
yy2 +
1
2mω2
zz2 (1.35)
with a mean trapping frequency ω = (ωxωyωz)1/3 and, according to equation 1.24, a
semiclassical (spatial and momentum) distribution function given by the occupation
number for single states
f (ϵ (r,p)) = f (r,p) =1
e(p2/2m+V (r)−µ)/(kBT ) + 1. (1.36)
Since the density of states in a harmonic trap is g (ϵ) = ϵ2/(2 (~ω)3), the atom num-
ber and total energy can be calculated using equations 1.27 and 1.28, respectively.
Similarly to equation 1.31 the spatial distribution becomes for T → 0 and µ0 = ϵF
n (r) =
∫d3p
(2π~)3f (r,p)
T→0→∫|p|<
√2m(ϵF−V (r))
d3p
(2π~)3. (1.37)
The Fermi momentum lies on a sphere of radius ~k (r) with unit spacing (2π~)3.The spatial density distribution of a zero temperature non-interacting Fermi gas is
thus
n (r) =1
6π2
[2m
~2(ϵF − V (r))
]3/2. (1.38)
The globally largest momentum is the Fermi momentum pF ≡ ~kF ≡√2mϵF . In
the Thomas-Fermi approximation (kBT ≫ ~ω), the Fermi momentum and energy
at each point of the trap r are approximated by a uniform gas such that
pF (r) = ~kF (r) =√
2mϵF (r) = ~(6π2n (r)
)1/3(1.39)
ϵF (r) = µ (r, T = 0) = ϵF − V (r) (1.40)
At T = 0, equation 1.27 and 1.28 may be written as
N↑ =
∫d3rn (r) =
1
6
( ϵF~ω
)3(1.41)
ϵF = ~ω (6N↑)1/3 = kBTF . (1.42)
22 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
From the condition that the profile vanishes for V (r) ≥ ϵF , the Fermi radius RF,i
in each direction i = x, y, z is calculated from V (RF,i) = ϵF as
RF,i =
√2ϵFmω2
i
. (1.43)
Explicitly, the distribution function reads
n (r) =8N↑
π2RF,xRF,yRF,z
[1−
(x
RF,x
)2
−(
y
RF,y
)2
−(
z
RF,z
)2]3/2
(1.44)
and the mean energy per particle is readily found from the ratio of equation 1.27
and 1.28 at T = 0,
E
N=
3
4ϵF . (1.45)
As per equation 1.29, for kBT ≫ ϵF (r) the gas shows a classical (Boltzmann) density
distribution n (r) ∝ exp(−V (r)/(kBT )). However, for kBT ≪ ϵF (r), the density
profile characteristically approaches n (r) ∝ (ϵF − V (r))3/2.
1.4.4 The interacting trapped case
Interactions are responsible for the pairing mechanisms between fermions of opposite
spin. In an ultracold gas with a balanced mixture of N↑ and N↓ fermions in two
different hyperfine states, interaction is allowed by lowest order collisions (s-wave
collisions). At T = 0, the Fermi momentum, kF , and the scattering length, a,
can be expressed as a dimensionless parameter, 1/(kFa), to quantify the strength
and sign of the interactions. They also determine the many-body properties. The
ground state in the BCS limit is reached for 1/(kFa) → −∞ because for weakly
attractive interactions composite fermions are always unstable against pairing but
simultaneously have to obey the Pauli principle; thus the density distribution is
close to an ideal Fermi gas. Weak repulsive interactions lead to the BEC limit,
1/(kFa) → +∞, where a tightly bound state exists to create molecules of mass
M = 2m. The Pauli principle is now a weak restriction on the bosonic character
and molecules can accumulate closer to each other. Maximum interaction strength
is reached for the unitarity limit, 1/(kFa) → 0, as the scattering length diverges
a→ ±∞ even at high temperatures. Here, the ground state is neither a BEC nor a
BCS because it is neither a bound state nor a Cooper pair defined. But the ground
state is always superfluid.
1.4 THERMODYNAMICS OF FERMI GASES 23
The following sections review the consequences on the density distributions and
thermodynamic quantities in the limiting cases. In general, the spatial distribution
of the clouds shrinks from the BCS side to BEC side due to the increasing interaction
strength.
1.4.4.1 BEC limit
When the scattering length is small and positive a > 0 and 1/(kFa) → +∞, fermions
will form bound dimers, see section 1.3.3, and acquire a bosonic character. These
dimers obey Bose-Einstein statistics. Thermal molecules can be cooled to molecular
Bose-Einstein condensates. For a trapping potential for molecules, VM (r), and
intermolecular interactions g = 4π~2aM/M , the many-body wave function Ψ (r) can
be described by the Gross-Pitaevskii equation(−~2∇2
2m+ VM (r) + g |Ψ(r, t)|2
)Ψ(r, t) = i~
∂
∂tΨ(r, t) . (1.46)
|Ψ(r, t)|2 is the condensate density nC , which for weak interactions and zero tem-
perature equals the density of molecules nM . For the ground state in equilibrium
one can use the ground state energy given by the molecular chemical potential µM .
The wave function is then replaced by Ψ (r, t) = exp (−iµM t/~)Ψ (r). When the in-
teractions dominate the kinetic energy of the condensate wave function, gnC ≫ ~ωi,
the Thomas-Fermi approximation yields the condensate density
nC (r) = |Ψ(r)|2 = max
(µM − VM (r)
g, 0
)(1.47)
which is also the order parameter. The chemical potential is again determined by
the molecule number NM = (N↑ +N↓) /2 =∫d3rnC (r). The trap for molecules is
twice as deep as for atoms, VM (r) = 2V (r), which changes the density profile to
nC (r) =15
8π
NM
RxRyRz
max
(1−
∑i
x2iR2
i
, 0
). (1.48)
The Thomas-Fermi radius is given by√
2µM/Mωi. Bose condensed clouds still show
a clear difference in the distribution compared to the thermal cloud.
1.4.4.2 BCS limit
In the BCS limit of weak attractive interactions, the scattering length is small and
negative a < 0 and 1/ ((kFa)) → −∞. Compared to the ideal gas case, the density
24 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
profile is marginally influenced by the interactions or pairing behaviour because the
sharp Fermi surface in k-space at kF is modified only in an exponentially narrow
region of width kF exp (−π/2kF |a|). The density distribution is of the form given
in equation 1.44 and the Fermi energy for N/2 = N↑ becomes ϵF = ~ω (6N↑)1/3 =
~ω (3N)1/3. Very low temperatures are necessary to observe the characteristic Fermi
profile. For this case (kF |a| ≪ 1), the many-body problem can be solved both at T =
0 and at finite temperature using BCS theory, which describes superconductivity in
metals [38]. The ground state is described by creation and annihilation operators
c†k,σ and ck,σ for fermions of momentum k and spin σ =↑, ↓
Ψ0 =∏k
(uk + vkc
†k,↑c−k, ↓
)†|0⟩ . (1.49)
uk and vk are the probability amplitudes. The chemical potential of the pairs in the
ground state and the temperature dependent pairing gap ∆ (T ) are the characteristic
quantities. The pairing gap is also the order parameter of the system. Since the
interactions have a small influence on the density distribution, the Thomas-Fermi
radius is taken for a non-interacting Fermi gas and compressed by a first-order
correction, so
Ri =
√2µ0
mω2i
= RF,i
(1− 256
315π2kF |a|
)(1.50)
where kF is the Fermi wave vector of a non-interacting gas. If kF |a| ≪ 1, then the
Thomas-Fermi radius becomes equal to the radius of a non-interacting gas
RF,i =
√2kBTFmω2
i
(1.51)
and the density distribution is given by
n (r) =4N
π2RxFR
yFR
zF
[1−
(x
RF,x
)2
−(
y
RF,y
)2
−(
z
RF,z
)2]3/2
(1.52)
The equation of state is approximated by that of an ideal Fermi gas, that is
µ (n) =~2
2m(3πn)2/3 . (1.53)
1.4 THERMODYNAMICS OF FERMI GASES 25
1.4.4.3 Unitarity regime
In the limit of strong interactions, the scattering length diverges a → ±∞ and
kF |a| → 0. It follows that the interparticle distance, l = n1/3 ∝ 1/kF , is the
only relevant length scale and the Fermi energy is the only relevant energy scale
ϵF = ~2k2F/(2m). For example, the scattering cross section maximises σ = 4π/k2.
Due to the independence of microscopic details, this model is applicable to any
other strongly interacting fermionic system where the scattering length diverges.
This property is called universality. The chemical potential reduces by a universal
parameter ξ = 1 + β, so µ = ξϵF = (1 + β) (~2/(2m)) (3πn)2/3. Using the local
density approximation, in which the gas is regarded as homogeneous in a small
sphere, the chemical potential is µ (r)− V (r). The density distribution in one spin
state follows as
n↑U (r) =1
6π2
(2m
ξ~2
)3/2
(µ− V (r))3/2 . (1.54)
Using the atom number integral
N↑ =
∫drrn↑U (r) =
1
6
(µ√ξ~ω
)3
(1.55)
the density profile is
n↑U (r) =8
π2
N↑
RUxRUyRUz
[max
(1−
∑i
x2iR2
Ui
, 0
)]3/2. (1.56)
This is identical to the density distribution of a non-interacting Fermi gas because
the equation of state µ ∝ n2/3 has the same power-law dependence. The universal
constant ξ simply rescales the radii (by a factor ξ1/4) and the central density (by a
factor ξ−3/4). One thus has direct experimental access to the universal constant ξ
by measuring the size of the cloud at unitarity although the universal constant has
only a very weak dependence on the radius. At a present understanding, an exact
solution of the many-body problem for kF |a| ≫ 1 is given by the relations outlined
in chapter 2.
1.4.5 Hydrodynamic expansion of clouds
For this thesis, ultracold atomic samples are prepared and imaged in-situ or after
release at and only at unitarity in the residual curvature of the high magnetic field. In
26 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
chapter 5, the imaging of atomic distributions at different temperatures is carefully
studied. The temperature of the coldest clouds are measured down to (0.09± 0.01)
T/TF , which is well below the critical temperature of 0.2 T/TF , creating therefore
superfluids. For increasing temperatures the normal phase contributes more and
more. For a gas in this regime the dynamics are well described by the Boltzmann
equation. But anisotropic expansion of a cold Fermi gas was measured at higher
temperatures (on the order T ≈ TF ), where the gas is not a superfluid, and found
to exhibit a similar hydrodynamic behaviour [37].
Hence, when unitary atomic distributions are released, one can assign hydro-
dynamic expansion laws. In general, these aspects can be very complex in different
crossover regime. Nevertheless, the discussion here is restricted to the unitarity
regimes, where the scattering length is large and the mean free path of the atoms
is on the order of the interatomic distances, 1/k ∼ l. The strong interactions raise
not only the critical temperature but also the critical velocity, compared to the lim-
its kFa → ±∞. Therefore, the hydrodynamic expansion laws can be extended to
higher temperatures at unitarity.
Hydrodynamic equations as well as the collisional and collisionless regimes are
usually discussed in the context of macroscopic phenomena characterised by long-
wave excitations, where collective oscillations are precisely measurable and exhibit
collective features unambiguously, e.g. see [92]. Hydrodynamic equations have been
used to describe the dynamics of BECs but it was suggested to extend them to
explain Fermi gases [113].
At T = 0, the macroscopic behaviour of a neutral superfluid is governed by the
Landau equations of irrotational hydrodynamics. The condition of irrotationality is
a consequence of the occurrence of off-diagonal long-range order, characterised by the
order parameter [92]. The hydrodynamic equations of superfluids consist of coupled
and closed equations for the density and the velocity field [114]. Derivations and
discussions on hydrodynamics of superfluids can be found elsewhere, e.g. [92, 112].
For the released clouds collisions during the expansion cannot be neglected
as for ballistic expansion. Hence, it is not possible here to extract the original
momentum distribution of the particles from the density distribution after the trap
is switched off. Instead, as the collision rate is higher than the largest trapping
frequency, clouds are imaged in the hydrodynamic regime. In fact, the expansion
occurs into an inhomogeneous magnetic field with a saddle potential
V (r, t > 0) =1
2m(ω2Sxx2 + ω2
Syy2 + ω2
Szz2)
(1.57)
1.5 SUMMARY 27
with real and imaginary frequencies ωSi. The chemical potential, µ, has the power-
law dependence on the superfluid density, n(r, t), thus µ ∝ nγ, where γ = 2/3 is
exact at unitarity and in the BCS limit. Hence the scaling ansatz
n (r, t) =1
bx (t) by (t) bz (t)n
(x
bx (t),
y
by (t),
z
bz (t), t = 0
)(1.58)
provides the exact solution. The scaling parameters bi follow the time-dependent
equations
bi = −ω2Sibi +
ω2i (0)
bibx (t) by (t) bz (t)(1.59)
which can be solved numerically. Here, ωi is the initial trapping frequency.
1.5 Summary
The study of strongly interacting Fermi gases forms a rich testbed to investigate pair
correlations. Previous experimental as well as theoretical work has paved the way to
understand new effects in unitary Fermi gases. Collisions in low temperature Fermi
gases are dominated by the two-body s-wave scattering length, a. In the vicinity
of a broad Feshbach resonance, the scattering length can be tuned via an external
magnetic field over a wide range. 6Li provides a very broad Feshbach resonance at
834 G and constitutes a good candidate to investigate the BEC-BCS crossover. The
unitarity regime is of special interest, since universal relations in dilute and strongly
interacting Fermi gases provide new, general insights into strongly correlated Fermi
systems. The interactions modify the thermodynamics of the Fermi gas dissimilarly
over the whole crossover region. The basic equations for the unitarity and ideal case
are introduced which are used for this thesis. Hydrodynamic equations characterise
the dynamic behaviour of unitarity-limited Fermi gases.
28 CHAPTER 1. BEC-BCS CROSSOVER PHYSICS
Chapter 2
Universal relations
The unitarity regime forms the physical setting for studying universal properties
of strongly interacting Fermi gases. The strong interactions make it difficult to
apply standard perturbation theories as it is difficult to define a small parameter.
New models are necessary leading to strong coupling theories or improved quantum
Monte-Carlo simulations. The Tan relations marked a major development in de-
scribing the resonance regime: the simplification of the problem created analytical
expressions and the characteristic contact parameter, C. Universal relations and
properties are valid throughout the BEC and BCS regimes as well as for zero and
finite temperatures, any spin imbalance and few- or many-body systems. In this
chapter, we revisit the problem of the unitarity regime and describe how the Tan
relations forge ahead.
2.1 Introduction
The superfluid transition in ultracold Fermi systems presupposes interparticle inter-
actions and strongly interacting Fermi gases facilitate the investigation of resonant
superfluidity. The interaction strength is quantified by the Fermi wave vector kF and
the two-body scattering length |a| and is considered to be strong when 1/kF |a| → 0
and weak when 1/kF |a| → ±∞. In the latter case, |a| represents a small pa-
rameter and standard perturbation theory is applicable. Unitary Fermi gases are
both dilute, that is, the range of the interatomic potential is much smaller than
the interparticle distance, and strongly interacting, because the magnitude of the
zero-energy scattering length |a| exceeds the interparticle distance. Elastic collisionsare responsible for the energy distribution giving the scattering length and sign a
key role in the many-body problem [14]. The question arose whether all strongly
interacting Fermi systems share the same features. Coming from nuclear physics
29
30 CHAPTER 2. UNIVERSAL RELATIONS
Heiselberg [115] compared Fermi systems with large scattering lengths in different
density regions to point out the relevance of dilute degenerate Fermi gases for nu-
clear and neutron star matter and, vice versa, the relevance of models for high energy
neutron stars to the first magnetically trapped Fermi gas [17]. Ho introduced the
term ‘universal properties’ for degenerate quantum gases [63] in the unitary limit
where the cross section reaches independently of |a| a maximum value 4π/k2, k be-
ing the relative momentum of the scattering atoms. In fact, this independence of
any features of atomic potentials poses a challenging many-body problem as there
are no small parameters anymore. Simplification is achieved by the so-called ‘uni-
versal hypothesis’, an assumption that the only relevant length scale in the ground
state at unitarity is the interparticle spacing n−1/3, which is related to the particle
density n. For unitarity limited, dilute Fermi gases (for which the interaction range
is finite as well as larger than interparticle distances) the microscopic details of the
interaction potential become indistinct and universality becomes a property of the
many-body system. This means in turn that all dilute systems with sufficiently
strong interaction behave identically except for a scaling parameter. Ho also consid-
ered universality for higher temperatures [116] inspired by the early work near the
Feshbach resonance. Ergo, universality is a property of strongly interacting Fermi
gases [63, 115, 117] and one can therefore study universal properties in ultracold
atomic gases to help understand other strongly interacting Fermi superfluids, e.g.
neutron matter [118, 119]. The principle of universality is studied in other research
areas, such as for phase transitions in general thermostatistics [120] or the β-decay
in nuclear physics [121–124].
Universal relations in ultracold Fermi gases represent a major breakthrough in
understanding highly correlated ensembles and their importance is to date only just
being established [125]. One of the first results on universality was the derivation
and measurement of the virial theorem for unitary Fermi gases together with a uni-
versal expression for the hydrodynamic expansion under isentropic conditions [49].
The Tan relations, being exact and widely applicable, had an avalanche effect on the
investigation of universal properties in strongly interacting Fermi gases [64–66]. Tan
exploited the fact that in the zero-range limit a well quantified parameter reciprocal
to the two-body interaction strength (scattering length), can enter the many-body
Schrodinger wave function of a non-interacting gas as a single, characteristic pa-
rameter. The so-called ‘contact’, C, specifies the likelihood of finding two composite
fermions close enough to interact with each other in a certain integration volume.
Stemming from the strong correlations, the number of pairs counted in this vol-
ume appears to be higher compared to the number derived from a downscaling of
2.1 INTRODUCTION 31
macroscopic considerations; hence, an enhancement of collective effects occurs due
to the energy carried by every pair. The contact encapsulates all of the information
required to determine the many-body properties and depends on the s-wave scatter-
ing length (interaction strength), spin compositions and temperature of the system.
The validity of the contact on a short interparticle range, 1/ |a| ≪ k ≪ 1/r0, im-
plies the effectiveness of the contact for large momenta and, furthermore, in the high
momentum limit of the density distribution of the gas.
Exact universal relations can shed light on new aspects of strongly interacting
Fermi systems and are relevant to research areas dealing with dilute Fermi sys-
tems with strong interactions. Usually, theoretical models for ideal cases can be
found [126]. However, when interactions are switched on, methods like perturbation
theory are necessary to describe the system [127–129]. The odd thing about the
unitarity regime is that on the one hand the interactions become too strong, such
that exactly these models, such as perturbation theory, fail. On the other, these
strong interactions and correlations cause effects which must lead to a nonpertur-
bative model (as everything is only depending on the scattering length and which
falls out of the problem) and hence analytic expressions become available again.
Triggered by the Tan relations, further theoretical work has strived towards
rederivations of some Tan relations within other theoretical models and derivations
of new universal relations [68, 71–73]. Within a quantum field theory, universal
relations follow from renormalisation and from the operator product expansion [67].
Such a framework allows the derivation of additional universal relations and the
systematic inclusion of corrections associated with the nonzero range of interactions.
Also, the ‘contact density’ appears in the short-distance expansion for the correlator
of the quantum field operators that create and annihilate the atoms.
In this chapter, the general idea of universal relations is discussed. It is neces-
sary to distinguish the contact for the homogeneous and trapped case. The relevant
definitions are introduced in section 2.2. The contact appears in a number of univer-
sal relations. However, in section 2.3, we revisit only a selection of relations, which
are the pair-correlation, the adiabatic sweep and the generalised virial theorem. Sec-
tion 2.4 discusses briefly the contact in the crossover region in regard of chapter 6. In
order to provide an understanding for the remainder of this thesis, different theories
on the universal contact of strongly interacting fermions at finite temperatures are
compared in section 2.5.
32 CHAPTER 2. UNIVERSAL RELATIONS
2.2 Definitions of the contact
Several presentations of the contact are circulating in the literature. The contact
can be locally defined, denoted as C(r), and is therefore sometimes called the contact
density. In a homogeneous system with volume V , one can use the average contact,
V C. Note that the contact of the uniform gas is also solely determined by the
equation of state, as experiments have shown [130]. From a comparison of the
prediction with the measured large momentum part, one could find out whether a
particular equation of state is correct. In this way, the experimental data in the
large momentum part give details about the equation of state of the uniform gas.
In order to predict V C for a trapped cloud, the local density approximation
(LDA) is used to calculate the local contact C(r) from the locally uniform gas. In a
trapped Fermi gas, the Thomas-Fermi approximation can be used to obtain locally
a uniform gas with enough atoms to satisfy the validity of the thermodynamic limit
and the thermodynamic quantities. Then, the local contact C (r) is still an exact
concept, the reason being that there is a negligible contribution to C (r) /k4 for a
small change of δk ≪ 1. Using the Thomas-Fermi approximation, the spatial contact
can be considered in a trapped gas as
V C =
∫C (r) d3r ≡ I. (2.1)
Note that if the momentum distribution nσ(k) for a spin state σ is normalised to
the particle number, Nσ =∫d3k/(2π)3nσ(k), then C is an extensive quantity with
dimensions of a momentum. It is therefore conventional to consider C in units of the
Fermi momentum, kF , divided by the total number of particles, N = N↑ +N↓. For
the remainder of this thesis, the notation of the contact reads for the homogeneous
and trapped case, respectively,
CnkF
(homogeneous) (2.2)
INkF
(trapped) . (2.3)
2.3 The contact and universal theorems
The contact enters several relations and properties for strongly interacting Fermi
gases. Some of those which have been experimentally accessible are discussed in the
following. In order to introduce the Tan relations, the physical scenario and the
2.3 THE CONTACT AND UNIVERSAL THEOREMS 33
0 1 2 3 4 50
0.5
1
1.5
k/kF
(k/k
F)4 n(k
) (a
.u.)
Figure 2.1: Visualisation ofextracting the contact. Sketchof the momentum distributionn(k) [131–133]. The contact fol-lows from C = limk→∞ k4nσ(k).The critical momentum is about2 k/kF .
relevant length scales are outlined first. A non-interacting two-component Fermi
gas is described by a wave vector k, spin σ =↑, ↓ and Fermi wave number kF . The
energy for a noninteracting gas, Enon−int =∑
kσ ϵknσ(k), depends on its momentum
distribution, nσ(k) ≡⟨c†σ (k) cσ (k)
⟩, which is built up from annihilation (creation)
operators cσ (k) on the momentum states per spin. The potential range of the
interaction is labelled r0 and the mean interparticle distance is denoted as l = n−1/3.
The Fermi gas in the BEC-BCS crossover has additionally a large scattering length
|a|. To study short-range physics, the characteristic length scale, r ∝ 1/k, must
be larger than the interatomic distance, r ≫ r0, but smaller than the other length
scales, i.e. the system is dilute r ≪ l, strongly interacting r ≪ |a| and ultracold
r ≪ λdB, where λdB is the deBroglie wave length; actually, r0 ≪ 1/k ≪ l ≪ λdB ≪a.
Equations 1.10 and 1.11 already pointed out constraints for the zero-range
limit. This model simplifies the description of universal properties of a general
model with strong interactions, since the scattering length is the only quantity that
arises from interactions. Despite this interpretation, the disadvantage of the zero-
range limitations are divergences of some observables, see below, and singularities
in intermediate steps of the derivations of the universal relations.
The zero-range model enters the Schrodinger equation for non-interacting par-
ticles by considering the boundary conditions given by equation 1.11. The N1 parti-
cles in state |↑⟩ andN2 particles in state |↓⟩ form a wave function, Ψ(r1, ..., rN1 ; r′1, ..., r
′N2),
which is according to the Pauli principle totally antisymmetric in the first N1 po-
sitions, r1, ..., rN1 , as well as the last N2 positions, r′1, ..., r′N2. This wave function
diverges for r1 = r′1 for any pair of fermions with different spin. When r1 and r′1 are
34 CHAPTER 2. UNIVERSAL RELATIONS
nearly equal, the wave function takes the form [125]
Ψ
(R+
1
2r, r2, ..., rN1 ;R− 1
2r, r′2, ..., r
′N2
)→ ϕ (r) Φ
(r2, ..., rN1 ; r
′2, ..., r
′N2;R)(2.4)
Φ(R) is a smooth function, where R is the centre of mass position of the pair. Note
that r is the general space coordinate and not assigned to any particle. ϕ (r) is the
zero-scattering wave function for two particles. In order to investigate correlations,
it is obvious to consider the relative wave function between composite spins, which
behaves like
ϕ↑↓ (r) =
(1
r− 1
a
)Φ (r2, ...,R) +O (r) . (2.5)
In the limit of closely spaced counterparts, i.e. short spacing r, the interaction is
understood as a contact interaction. The resulting interaction energy, Eia, due to
the large scattering length modifies the total energy of the system. The expectation
value of the total energy of the system is the sum of the internal and potential
energy, Etot = Eint + Epot = Eia + Ekin + Epot. Both the kinetic energy, Ekin, and
the interparticle interaction, Eia, depend each sensitively on the short scale physics,
or equivalently their large momentum physics k → ∞ in the limit of r0 → 0. It is
necessary to cut off at a range such that the kinetic energy Ekin behaves ∝ 1/r0 for
r0 → 0. Then, the momentum distribution decays like 1/k4 at large k.
Tan reformulated the integral representation of this famous s-wave contact
interaction problem to relate the energy and momentum distribution [64] to the
contact. Tan showed that the sum of the two energies (Eint = Ekin + Eia) is inde-
pendent of the details of the short-distance physics. The simple relation between
the energy and the momentum distribution, where the only microscopic parameter
is the scattering length a, may be written as
Eint = Eia + Ekin =~2V C4πam
+ limK→∞
∑k<K,σ
~2k2
2m
(nσ (k)−
Ck4
). (2.6)
The summation over momentum is∑
k ≡ V∫d3k/(2π)3. The contact is defined as
C ≡ limk→∞
k4n↓ (k) ≡ limk→∞
k4n↑ (k) . (2.7)
The mathematical proof is elegant but lengthy and necessitates functional expres-
sions much like generalised delta distributions. Therefore the interested reader is
referred to the original literature [64] or the review article [125].
2.3 THE CONTACT AND UNIVERSAL THEOREMS 35
Both equations 2.6 and 2.7 are Tan relations. In equation 2.7, C is the same
for both spin states. Equation 2.6 holds for any finite-energy states, few-body or
many-body, equilibrium or nonequilibrium, zero temperature or finite temperature,
superfluid state or normal state, and any (im)balanced spin population.
2.3.1 Pair-correlation function
In this thesis, the contact is experimentally determined via the definition of the
short-range pair correlation function between fermions with opposite spins. Details
on the measurement of the pair-correlation function and the contact can be found in
chapters 6 and 7 of this thesis. The link to the pair correlations is probably the most
intuitive interpretation of the contact. Taking equation 2.4 for the wave function
for one fermionic pair of composite spin, one can express the spin-antiparallel pair
correlation function as
g(2)↑↓ (R) ≡
∫dR ⟨n↑ (R− r/2) n↓ (R+ r/2)⟩ . (2.8)
where n↑ and n↓ are the number densities. Short-range (R → 0) structure in a
quantum fluid depends upon the relative wave function of the interacting fermions
in different spin states, ϕ↑↓(r) ∝ 1/r− 1/a, see equation 2.5. The contact parameter
appears as a prefactor
g(2)↑↓ (R → 0) ∼=
C16π2
(1
r2− 2
ar
)(2.9)
at small r. This means that the correlation between the number densities for the
two spin states at points separated by a small distance r diverge as 1/r2 and the
coefficient of the divergence is proportional to the contact. Sometimes it is conve-
nient to work in Fourier space, for instance, when probing the correlation function
with Bragg spectroscopy. The Fourier transform of equation 2.9 is
ϕ (k) =4π
k2− (2π)2
aδ3 (k) (2.10)
From equation 2.1 and 2.9 it follows that the expectation of the number of pairs of
fermions with diameters smaller than a small distance s≫ r0 or in a certain volume
element d3r is
dNPair =C (r) s
4πd3r → s4
4C (r) . (2.11)
36 CHAPTER 2. UNIVERSAL RELATIONS
The volume is V = (4π/3)s3 and the number of pairs is Npair = N↑N↓. Hence, the
contact measures the number of pairs and fermions with different spin states that
have a small separation. The s4-scaling has a so-called anomalous scaling dimension,
as it is not s6, due to the strong correlations. The contact density corresponds to
the local pair density when the range of interactions is similar to the interparticle
distance, 1/k ∼ l. This is not to be confused with Cooper pairs which are totally
dominated by a correlation of momentum, 1/k ≫ l.
2.3.2 Adiabatic sweep theorem
Another way the contact can be determined is via the adiabatic sweep theorem. The
high-momentum tail of the momentum distribution of the gas determines the number
of large momentum fermions (on average), see equation 2.7, Nk→∞ = V C/π2k plus
higher order corrections which are negligible at large k. By tuning the inverse
scattering length adiabatically while keeping the confinement potential fixed, the
total energy E of the gas will change. The adiabatic derivative of energy, expressed
as dE/d(1/a), and the contact for the magnitude of the 1/k4 tail are two independent
quantities. Tan derived the simple, exact, and universal relation [65]
~2V C4πm
=dE
d (−1/a)(2.12)
which is called the adiabatic sweep theorem. The rate of change of energy due to a
small change of the inverse scattering length is proportional to the contact, for fixed
entropy and particle numbers. The definitive form of equation 2.12 is not obvious. In
other words, equation 2.12 gives the magnitude of the large momentum part, i.e. the
contact density C, when the derivative of the energy with respect to −1/a is taken.
A small and slow change in the reciprocal of the scattering length is equivalently
treated with quantum mechanical first-order perturbation theory which determines
the amount of energy level shift. In the limit of a zero effective interaction radius,
two fermions in opposite spin states only interact if they appear at the same position,
so the energy shift should be proportional to the probability that this occurs. Then
their momenta are large and nearly opposite and V C a characteristic quantity. If
one substitutes the energy with the free energy F = E − TS in equation 2.12,
then this expression determines the thermodynamics of the system, as F enters all
other thermodynamic functions. When changing the scattering length suddenly, the
energy shift is also proportional to the contact. Tan also proposed that a sudden
change in the scattering length could help to detect dissociation of a fermionic dimer.
2.4 CONTACT IN THE BEC-BCS CROSSOVER 37
2.3.3 Generalised virial theorem
A virial theorem has been shown in the unitarity limit stating that the total energy
is twice the potential energy E = 2Epot, but theoretical calculations were restricted
to 1/(kFa) ≡ 0 and the local density approximation [49]. In chapter 5, this relation
is used to calculate the temperature from the energy per particle at unitarity. Tan
generalised the virial theorem to finite scattering lengths. Let us consider the general
confinement potential V (r) = rβf (r) where β > −2, β = 0 and βf (r) > 0. f (r) is
a smooth function. Then the generalised virial theorem is
Etot −β + 2
2Epot = − ~2C
8πam. (2.13)
For harmonic trapping potentials, (m/2)ω2r2, it follows β = 2 and
Ekin + Eia − Epot = − ~2C8πam
(2.14)
since Etot = Ekin + Eia + Epot. For a small and negative scattering length, a→ 0−,
as in the deep BCS limit, the ratio behaves like C ∝ a2 and the virial theorem for
the noninteracting Fermi gas is recovered. The unitarity case, a → ∞, and a trap
confinement β = 2 lead to E = 2Epot. For large atom numbers N↓ = N↑ ≫ 1
and a small and positive scattering length, a → 0+, the system approaches in the
zero temperature limit a BEC of tightly bound molecules, so the above expression
approaches the virial theorem for the Gross-Pitaevskii equation for these bosonic
molecules [134] with scattering length am ≈ 0.6a [27]. In equation 2.13 the details
of the statistical distribution of the energy levels are unimportant, as long as the
statistical weight decays sufficiently fast at large energy and the average value of Cfor the individual energy levels can be applied. Thus, this expression is valid for
any finite temperature states in the canonical or grand canonical ensemble, as well
as the ground state.
2.4 Contact in the BEC-BCS crossover
The contact parameter plays a part in setting the many-body properties and ther-
modynamic quantities in the phase diagram, as shown in figure 1.1. In the zero
temperature limit, the contact has been calculated [84, 131, 132, 135] and measured
with several experimental techniques [70, 133, 136], as shown in figure 6.7a in chap-
ter 6. The contact will be adequate in the crossover region if it recovers the known
results in both the BEC and BCS limits. The analytic expression of the ground
38 CHAPTER 2. UNIVERSAL RELATIONS
state energy is known in these limits. Using for instance the adiabatic sweep theo-
rem equation 2.12, the contact is definable. In the following, the particle density is
denoted as n = k3F/(3π2).
Contact in the BEC limit
In the BEC limit the ground state is expanded in terms of the Bose gas parameter
nda3dd [137–139,139,140], where the dimer-dimer scattering length is given by add =
0.6a [111]
E
V= ndEd +
2π~2n2dadd
md
(1 +
128
15√π
√nda3dd + ...
)(2.15)
where nd/2 is the dimer density and md = 2m is the dimer mass. Ed = −~2/(ma2)is the energy of an isolated dimer at rest. The second term considers the mean-field
energy of a BEC of dimers. For 1/(kFa) → +∞ the contact becomes C → 4πn/a
from equation 2.15 and 2.12 .
Contact in the unitarity limit
In the unitarity limit, the expansion is [118,141,142]
E
V=
3
5
~2k2Fn2m
(ξ − ζ
kFa− 5ν
3 (kFa)2 + ...
). (2.16)
The universal numerical constant, ξ ≈ 0.42± 0.01 [130,143], is used in this work to
calculate the true temperature. The parameters ζ ≈ 1 and ν ≈ 1 were originally
extracted with Quantum Monte-Carlo calculations [142,144,145], but more recently
ζ was measured to be ζ ≈ 0.91 [130]. Also, ζ ≈ 0.95 from equation 2.9 [125] and
ξ ≈ 0.383(1), ζ ≈ 0.901(2), ν ≈ 0.49(2) from QMC calculations [131]. After applying
equation 2.16 and 2.12 the contact is of the form C → k4F2ζ/(5π). This dependence
on k4F is on dimensional grounds as the interaction does not provide a length scale
at 1/(kFa) = 0.
Contact in the BCS limit
In the BCS limit, the ground state energy is expandable in terms of the interaction
parameter 1/(kFa) and may be written as [90,118,142,143,146]
E
V=
3
5ϵFn
(1 + vBCS,1 (kFa)
1 + vBCS,2 (kFa)2 + vBCS,3 (kFa)
3 + ...)
(2.17)
2.4 CONTACT IN THE BEC-BCS CROSSOVER 39
where the coefficients vBCS,1 = 10/(9π) and vBCS,2 = 4 (11− 2 ln 2) /(21π2) are
exact [137] but vBCS,3 = 0.030(5) is numerical [147]. Consequently, the contact
follows from equations 2.17 and 2.12 and approaches in the BCS limit C → 4π2n2a2
and the contact density decreases to 0 as a→ 0− with an a2-dependence. In contrast,
the BCS superfluid energy has an exponential character, ∝ exp (−π/2kF |a|) for
a → 0−. While Cooper pairs can be explained with standard BCS equations, the
BCS theory does not predict the correct value of C in the BCS limit [65].
Verifications of the contact in the BEC-BCS crossover
The contact can be inferred from different definitions, which makes the contact
so powerful and reflects its versatility, and its values can be taken from different
kinds of experiments. Experimental verifications showed the monotonic decrease
of the contact from the BEC to BCS limit. In this work, Bragg spectroscopy is
used to extract the behaviour of the pair-correlation function at high momenta,
[26, 75,80,148–150] and is presented in chapter 6.
The contact was first extracted with photo-association [70]. The excitation
rate Γ = Ω2R/γNmol depends on in the Rabi frequency ΩR, the line width γ and the
number of photo-associated molecules Nmol.
The contact is directly accessible by using equation 2.7 and measuring the large
momentum tail of the radial momentum distribution n(k) [133]. The external in-
teractions are switched off through fast ramps of the magnetic field such that the
scattering length vanishes. The radial momentum distribution is taken from an ab-
sorption image of a ballistically expanded cloud and an Abel transform. Then, the
momentum distribution is totally determined by the kinetic energy. The measure-
ment will exhibit a curve like that shown in figure 2.1. While the cloud is always
imaged at zero-scattering length, the cloud can be prepared at different initial scat-
tering lengths (magnetic fields); therefore this method allows the contact to be
measured across the BEC-BCS crossover. This method depends on the speed of the
magnetic field ramp to zero-scattering length and good image qualities.
Radio-frequency (rf) and photo-emission spectroscopy (PES) measure the bulk
spectral function of an ultracold gas. The rf-transition rate Γ(ν) for an excita-
tion frequency ν shows at high frequencies a ν−3/2-dependence such that Γ(ν) →C/(23/2π2ν3/2) [133, 151]. The rf-transition between two (hyperfine) states selects
the contact between atoms in these two (hyperfine) states, given final state effects
are small. Similarly, PES can also couple out atoms from a certain state [133].
Angle-resolved PES (ARPES) can additionally measure excitation spectra depend-
40 CHAPTER 2. UNIVERSAL RELATIONS
ing on the transferred momentum, such that one obtains information about the
total momentum distribution n(k, ν). After frequency integration and using again
equation 2.7, the contact can be extracted.
Equation 2.12 has also been measured since adiabatic sweeps are controllable
with adiabatic magnetic field ramps [133]. Tan suggested an experimental sequence
to directly test the adiabatic sweep theorem in a trap at any temperature [64].
Also, in an adiabatic sweep the entropy is conserved. The experimental verification
is reported in [133]. The total energy of a cloud can be measured with the mean
square size, σ2, Eint = Ekin +Eia = (2/ϵF )(m/2)ω2σ2 in terms of the Fermi energy,
ϵF . By measuring the mean square size with and without the external potential and
comparing images at zero- and finite scattering length, the contact can be extracted
from equation 2.14. For different starting scattering lengths, the contact across the
BEC-BCS crossover is obtained.
The homogeneous contact may be extracted from the equation of state [130].
2.5 Universal contact at finite temperatures
The contact is an important quantity for characterising many-body properties and
phases in strongly interacting Fermi gases. The contact C depends on the tempera-
ture T/TF (relative to the Fermi temperature TF = ~2k2F/(2mkB)) and can be used
to study the complete momentum distribution in the entire (−1/(kFa), T/TF )-plane
(for balanced populations of the two spin states). There remain open questions
about the details of the nature of unitary-limited Fermi gases and measurements of
the contact may help to clarify the mysterious pseudogap. The determination of the
temperature of a strongly interacting Fermi gas is complicated and will be discussed
in chapter 5. In the next sections, the contact is briefly reviewed in relation to finite
temperatures, because the experimental results are compared to the different calcu-
lations for the trapped case in chapter 7. Different theoretical models describe the
zero-temperature coupling as well as the temperature dependent contact at unitarity,
which are characterised by the contact, in the temperature-coupling phase diagram.
The contact parameter describes how the short-range (high-momentum) physics im-
pacts on the bulk properties of the system. In contrast, it is not clear if the contact
can be related to the low-energy properties of the system, where so far the so-called
‘Bertsch parameter’ and the pairing gap are characteristic features [92, 152, 153].
Although only the trapped case is measured in this work, it is necessary to mention
also the homogeneous case to understand where the physical signatures find their
origin. The accuracy of the theoretical models can be quantified by their predictions
2.5 UNIVERSAL CONTACT AT FINITE TEMPERATURES 41
for the contact.
2.5.1 Thermodynamic contact from the entropy
Tan suggested the determination of the temperature in the unitarity regime with
the frequently used projection of the temperature to the weakly interacting regime
(BCS side) and then using the adiabatic sweep theorem, equation 2.12 [65]. The
entropy is preferred over the reduced temperature T/TF due to its invariance under
adiabatic sweeps. Since the contact depends on the temperature, the determination
of the contact from the entropy and inverse scattering length would represent an
indirect temperature measurement. Starting from the BCS side and tuning the
inverse scattering length adiabatically into unitarity regime, or vice versa, one knows
the entropy of the final state from that of the well-known initial state. For various
initial entropies the derivative of energy with respect to entropy can be calculated at
any scattering length to obtain the temperature of the gas [65]. This principle has the
advantage of being model-independent, since almost no theoretical approximations
are involved.
2.5.2 Contact for the trapped case at zero temperature
The trapped contact at unitarity takes an analytic expression at T = 0 [135](I
NkF
)=
256
105πξ−1/4
(CnkF
)=
512ζ
175ξ1/4. (2.18)
because the density profile, n(r), is exactly known. The two universal parameters
ξ and ζ are related to the expansion of the ground state energy per volume of an
interacting Fermi gas at unitarity, see equation 2.16. They were measured to be
ξ ∼= 0.41(1) and ζ ∼= 0.93(5) in the ENS group [130], however ζ ∼= 0.90(1) from a
Quantum Monte Carlo simulation [131].
2.5.3 Below the critical temperature
Various strong coupling theories can be used to model strongly interacting Fermi
gases beyond standard perturbative methods. The given strong interactions require
assumptions in the fundamental equations (many-body Schrodinger equation, s-wave
contact problem), which reduce the theoretical model of choice. The Feynman-
diagrammatic structure in the t-matrix approximations needs an infinite ladder sum
to solve the principal equations, which are the t-matrix, particle-particle propagator
42 CHAPTER 2. UNIVERSAL RELATIONS
and self-energy, depending on the Green’s function [154, 155]. These equations can
be self-consistent (GG) [156–158], or non-self-consistent (G0G0) [159–161]. The
Noziere-Schmitt-Rink procedure uses additional path integrals to include Gaus-
sian (pair) fluctuations and is in the following referred to as the NSR or GPF
method [83, 146, 162–164]. While this thesis was being written it was discussed
which of the strong coupling theories is the most appropriate. The absence of a
small parameter and the possibility to choose a particular approximation eventually
prevent a common agreement on the prediction for the contact, especially in the crit-
ical temperature region. Details of these techniques can be found in [135]. Another
possibility is provided by quantum Monte-Carlo simulations (QMC) [131,132].
Homogeneous case
Due to the different approximations in the strong coupling models, the predictions
deviate slightly and show discrepancies around the critical temperature, especially
in the calculations for the homogeneous case. However, the result influences its
interpretation. The entropy relation(∂I∂T
)=
4πm
~2
[∂S
∂ (−1/as)
]T,µ
(2.19)
allows the contact to be directly obtained by taking the temperature derivative in
equation 2.19 [165]. Similar to the suggestion of Tan, the temperature dependence
of the contact is determined by the variation in the entropy with respect to the cou-
pling constant. It has been argued that at low temperatures, phonon excitations con-
tribute significantly with a maximum at ∼ (0.5±0.1) T/TF , using the G0G0-method
and ab-initio G0G0-method over the whole temperature range [84, 85, 165]. The in-
terpretation is that the local order is established by pairing fluctuations around the
critical temperature [84]. The contact is enhanced where pseudogap phenomena are
maximal. Fluctuations above TC strengthen the local pairing correlations in the ab-
sence of the long-range order. However, the contact is proportional to the two-body
correlation function, which is affected by pairing fluctuations in the particle-particle
channel. The definitions for the short-range boundary conditions need to be refined,
i.e. to identify the approximate spatial boundary between short-range (r ≪ k−1F ) and
medium-range (r ∼= k−1F ) physics. The characteristic moment kc is introduced, such
that 1/kc is the spatial range at which two-body physics interfaces with medium-
range physics, and is of the size of the composite boson. It is an open question how
the values of the contact are influenced by this medium-range physics and how the
presence of a pseudogap connects to the single-particle excitation spectrum.
2.5 UNIVERSAL CONTACT AT FINITE TEMPERATURES 43
In the NSR (GPF) method the contact decreases monotonically without a
maximum due to the break down of the strong coupling theories around the critical
temperature [135, 166, 167]. Within the GPF-theory, the phonon excitations con-
tribute in a temperature window T < 0.07 TF ≪ TC [135]. Fermi liquid theory
predicts a maximum, but its applicability is doubtful in the relevant short length
scale [135].
In comparison with strong coupling theories, QMC simulations offer another
way to access the contact [131,132]. They seem to be especially useful due to their
precision and freedom from uncontrolled approximations. One can derive Tan’s
expression for the contact parameter using just the short-range behaviour of the
ground-state many-body wave function and calculate the momentum distribution
and contact density. Within QMC methods, extracting the contact from the equa-
tion of state is the simplest and most reliable way [131]. QMC simulations for the
momentum distribution at different temperatures can also show that the contact
may have a maximum at ∼ 0.4 T/TF supporting the phononic scenario where the
contact increases with T/TF . However, problems may arise from low density edge
effects in a trapped cloud.
All theories predict that the contact decays slowly at higher temperatures be-
cause it is weakly depending on the long-range order. Measuring the spatial contact
in the homogeneous case provides more information compared to the contact in the
trapped case, where the different shells in the energy distribution have different lo-
cal T/TF . The result found for the universal homogeneous contact calculated with
different approaches from strong-coupling theories can be briefly summarised: With
the GPF theory C/(nkF ) = 3.34 was found whereas C/(nkF ) = 3.02 followed from
GG calculation [168] and C/(nkF ) = 3.23 resulted from the G0G0 theory [84]. Ex-
perimental investigations showed C/(nkF ) = 3.51 ± 0.19 from the equation of state
data at ENS [130]. QMC calculations have yielded C/(nkF ) = 3.40 [75], C/(nkF ) =3.39(6) [131] and C/(nkF ) = 2.95± 0.1 [132].
Trapped case
Compared to the homogeneous case, the contact is translated into the trapped case
with a weighting of the energy shells, n(k)4/3. Experimental values for I/(NkF )were found to be 3.4 ± 0.18 (ENS) and 3.1 ± 0.3 (this work). The theoretical pre-
dictions for I/(NkF ) are 3.26 (GPF), 3.03 (GG) and 3.05 (G0G0). Trap averaging
washes out the maximum just below TC [84, 85] and brings all predictions in qual-
itative agreement over the whole temperature range. The contact for an isotropic
trap is derived from equation 2.1 and the LDA. The Fermi momentum is replaced by
44 CHAPTER 2. UNIVERSAL RELATIONS
k3F = n(r)3π2 and the density, n(r), depends on the reduced temperature, T/TF (r).
The integral to obtain I/(NkF ) is cumbersome and can be solved in different nu-
merical ways. Due to the particular calculations with strong-coupling theories at
low temperatures, T < 0.3 TF , the universal trapped contact spreads over the values
mentioned above.
2.5.4 Above the critical temperature
Above the critical temperature the fugacity is a small parameter, z ≡ exp (µ/(kBT )),
and therefore quantities can be expanded as a function of the fugacity or inverse fu-
gacity. The quantum virial expansion (QVE) has been shown to accurately describe
the high-temperature range in strongly interacting Fermi gases [116, 149, 169–171].
For the contact calculations, either one can insert the expression of the tempera-
ture [165] into the QVE, or one can develop, analogously to the QVE of thermo-
dynamic potentials [169], the contact in terms of universal temperature indepen-
dent coefficients as a consequence of fermionic universality [135]. Thus the contact
evolves in powers of the fugacity like C ∝ [c2z2 + ...+ cjz
j + ...]. The contact co-
efficients, cj = ∂∆bj/∂(λdB/a), include the j-th virial coefficient, ∆bj, and the
thermal deBroglie wave length λdB. The first virial coefficients were predicted to
be ∆b2,∞ = 1/√2 [172] and ∆b3,∞ ∼= −0.355 [169] and confirmed by the ENS
group [173].
Homogeneous case
In the homogeneous case, the contact-QVE is valid down to T ∼ TF and one finds
c2,∞ = 1/π and c3,∞ = −0.141 [135]. These enter the high-temperature dimension-
less contact for the homogeneous gas as
CnkF
= 3π2
(T
TF
)2 [c2,∞z
2 + c3,∞z3 + ...
](2.20)
where the fugacity z is determined by a number equation [171].
Trapped case
For an isotropic trap the contact-QVE works for temperatures down to 0.5 T/TF
because the contact coefficients are by a factor of j3/2 smaller than for harmonic
traps, cj,∞,T = cj,∞/j3/2. As discussed before, for a harmonic trapping potential
VT (r) the LDA can be applied. This modifies the chemical potential µ(r) = µ−VT (r)and the local fugacity z(r) = exp(µ(r)/(kBT )) ≡ z exp(−VT/(kBT )). The contact
2.6 SUMMARY 45
coefficients are determined to be c2,∞,T∼= 1/(2
√2π) and c3,∞,T
∼= −0.0271 [135].
The dimensionless contact depends on these as
INkF
= 24π3/2
(T
TF
)7/2 [c2,∞,T z
2 + c3,∞,T z3 + ...
]. (2.21)
This expression explicitly shows that the contact in harmonic traps decays much
faster than in the homogeneous case. The reason is the reduction of the peak
density at the trap centre at high temperatures.
2.6 Summary
In this chapter the problem of unitarity limited Fermi gases has been introduced.
The lack of a small parameter complicates standard perturbation theories in this
strongly interacting regime. Universal relations overcome this problem by examining
the short-range structure, which is equivalent to characterising the problem in the
high-momentum limit. Many important relations and thermodynamic quantities
are related to the central parameter, the contact C. For the spin-balanced gas, it
is possible to measure the interaction and temperature dependence of the contact
and compare the data to numerical and analytical models. Contact measurements
are complementary to the measurements of the equation of state. The contact is
proportional to the observed energy, and therefore it is a more sensitive test of
theoretical predictions than a direct measurement of the thermodynamic quantity
[135]. Although there has been theoretical work on the temperature dependent
contact, discrepancies at the critical temperatures need to be resolved by accurate
experiments. Possible candidates are measurements of the homogeneous equation of
state near unitarity, measurements of the tail of the spatially-resolved rf spectrum
at unitarity, and measurements of the spatially-resolved Bragg scattering spectrum
at unitarity.
46 CHAPTER 2. UNIVERSAL RELATIONS
Chapter 3
Structure factor of a Fermi gas
The strong correlations found in interacting Fermi gases at unitarity naturally lead
one to the study of the spectrum of elementary excitations. In a measurement, they
will respond linearly to an external probe if the interaction with the system is weak
and the system behaves as if it was undepleted. In general, inelastic scattering of
particles is a well established means of investigating correlation functions. Different
energy ranges can be covered: in solid state physics, fast electrons are used to study
charged Fermi liquids, inelastic scattering of slow neutrons measures correlations in4He and superconductors, whereas Bragg scattering of photons can probe correla-
tions in ultracold atomic quantum gases. In this chapter the theory and analysis
of inelastic Bragg scattering is outlined to form the physical background for the
experiments explained in detail later in the thesis.
3.1 Introduction
Linear response theory allows many-body problems to be addressed for interacting
systems at all temperatures as long as the probe-coupling interaction is weak. The
probe-system coupling can be treated as a weak perturbation and therefore many
quantities can be derived accurately from first order perturbation theory, which ex-
actly is the linear response of the system. Bragg scattering in the low momentum
limit can reveal the collective behaviour of the system through macroscopic exci-
tations. Excitations at high momenta reveal the momentum distribution or single-
particle excitations of the system. A detailed discussion on the differences of high
and low momentum Bragg spectroscopy of Bose-Einstein condensates is presented
in [174], where the Bogoliubov theory is applied to explain the excitation spectra.
Essential in linear response theory is the dynamic structure factor, S(q, ω), which
encapsulates all information about correlations and obeys particle conservation as
47
48 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
well as sum rules. To measure S(q, ω), linear response theory links it to a response
function containing the system properties and therefore S(q, ω) is a measurable
quantity that probes the spectral properties of the system. The static structure
factor, S(k), gives the net momentum transfer for the energy unresolved dynamic
structure factor. It represents the Fourier transform of the two-body correlation
function [175]. Linear response theory covers a wide range of scenarios and detailed
discussions can be found in various textbooks [1, 76]. However, for this thesis, it
is sufficient to consider density-density correlations, i.e. the measured density pro-
file is the response to a periodic perturbation of the probe density profile. Only
the limit where the transferred momentum is much larger than the Fermi momen-
tum |k| ≫ kF will be discussed here. Not included in these explanations is the
low momentum limit which could disclose superfluid mechanisms due to collective
behaviour.
It is common to define the dynamic structure factor either in terms of the
momentum and energy, such that S(q, E), or in terms of the wave vector and fre-
quency, thus S(k, ω). In this chapter, a mixed notation is used due to convenience
and the dynamic structure factor consists of the momentum and the frequency,
S(q, ω), while ~ is inserted in the equations if necessary.
Beginning with the motivation of Bragg spectroscopy as a tool to investigate
pair correlations, the principles and advantages are discussed in section 3.2. The
spectral functions are related to the dynamic structure factor, as shown in section 3.3.
Derived from these is the static structure factor, section 3.4, which is linked to
the pair-correlations. In section 3.5 it will be pointed out that sum-rules provide
a powerful constraint on the spectral function. The link to the linear response
function is made in section 3.6 and the response at finite temperature is discussed
in section 3.7, introducing the detailed balancing which arises from the fluctuation-
dissipation theorem. An explanation on the acquisition of S(q, ω) and S(q) from
the measurements is the subject of section 3.8.
3.2 Bragg spectroscopy of pair-correlations
A direct measurement of pair-correlations in ultracold Fermi gases can be difficult
because the spatial range of interest is usually much smaller than the resolution
of typical imaging systems. Bragg spectroscopy offers a model-independent tool
to probe pair-correlations in momentum space. The probe particles are photons
provided by the Bragg laser beams. The total momentum and the energy of the
photons have to be conserved in a photon absorption-emission process, where an
3.2 BRAGG SPECTROSCOPY OF PAIR-CORRELATIONS 49
atom compensates the momentum and energy transfer. By measuring the change
in the centre of mass or the energy of the atomic distribution after a short Bragg
pulse, the spectral response becomes evident. This leads to the dynamic structure
factor and the static structure factor, which is linked to the two-body correlation
function and the contact, C, see chapters 2 and 6.
3.2.1 Principle of Bragg scattering
An ultracold quantum gas is illuminated with two low-power Bragg beams that
intersect at an angle, θ, and form a moving periodic potential. The wave vector,
kL = 2π/λBr, of the laser depends on the wave length, λBr. Absorption of N photons
from one laser beam and subsequent stimulated emission into the other imparts a
momentum to a number of atoms where the order of the scattering process is N .
The momentum transfer may be written as
q = ~k = 2N~kL sin
(θ
2
). (3.1)
The energy transfer to an atom with an initial momentum kini is
~ω ≡ En − E0 =q2
2m+
~kini · qm
, (3.2)
where ω denotes the frequency difference between the two Bragg beams. These two
equations fulfil energy and momentum conservation. A resonance will occur when
the momentum of the moving potential, 2~k, matches the momentum of the atom
mv with mass m. For single atoms this simplifies to the usual Bragg resonance
frequency, ωR = 2~k2/m. The second term in equation 3.2 is related to the Doppler
shift of the resonance and shows that Bragg scattering is a velocity-selective process.
Energy and momentum can be decoupled if the angle between kini and q is small.
The energy is varied by the frequency difference, ω, between one beam, c/λBr, and
the other, (c/λBr)− ω, where c is the speed of light. The Bragg scattering process
is a two-photon process where fluctuations are caused by the time-varying electric
field of the Bragg laser beams. Hence, the potential seen by the atoms oscillates
with time and induces transitions of energy ~ω.The spatial period of the Bragg potential sets the momentum range being
probed. In spin-balanced fermionic gases, two fermions may scatter as a single
particle with mass 2m when the size of the pair is small compared to 1/k, where
k = |k|. Since the momentum range is freely selected, the relative probe momentum
k/kF > 2 allows the fraction of pairs to be mapped within a particular size range
50 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
0 < r < λBr. A sketch of this scenario is presented in figure 3.1 in an energy-
momentum diagram.
Figure 3.1: Principle of Bragg scattering sketched as energy versus momentum.The energy of the two-photon process between the ground and excited state lieson a q2-parabola. In 6Li, it is the transition
∣∣22S1/2, F = 1/2,mF = −1/2⟩
→∣∣22P1/2, F = 3/2,mF = −1/2⟩. The detuning ∆ from the upper level is necessary
to avoid spontaneous scattering. For a given momentum transfer, q, there exists aresonance frequency ωR = 2~k2/m. The laser frequency of the Bragg beams, c/λBr,is given by the speed of light, c, and the Bragg wave length, λBr.
3.2.2 Advantages of Bragg spectroscopy
The concept of inelastic scattering is a traditional technique to probe many-body
quantum systems [76], such as inelastic neutron scattering in superfluid helium or
inelastic light scattering from ultracold Bose gases [176–179]. After the first steps
in manipulating atomic samples with Bragg spectroscopy [180–182], its potential
for the investigation of properties of ultracold quantum gases has been slowly ap-
preciated [183]. In Bragg spectroscopy, two momentum states of the same internal
atomic state are connected by a stimulated two-photon process: The stimulated
process implies higher resolution and sensitivity in the measurement as the signal
is background free. Compared to radio-frequency spectroscopy [56, 86] or Raman
spectroscopy, Bragg scattering does not involve a third atomic state or an inter-
3.3 DYNAMIC STRUCTURE FACTOR 51
mediate metastable state and thus final state interaction effects are avoided. The
momentum range is broad and easily selected for fermionic gases. The momentum
transfer q = ~k and energy transfer ~ω are predetermined by the laser beams and
not postevaluated from the measurements. Bragg spectroscopy can probe density
fluctuations and measures S(q) which probes the two-body correlation function in a
many-body system. Coherent properties and S(q, ω) have been measured in atomic
Bose-Einstein condensates [26, 77] as well as the static structure factor [78, 79]. In
the fermionic case, Bragg spectroscopy has been successfully used to determine the
dynamic and static structure factors in the BEC-BCS crossover at a single momen-
tum [80]. Bragg spectroscopy of atoms confined in periodic lattices aims to find
effects similar to those found in solid state physics, for instance band structures and
transitions from superfluid to Mott-insulator states [184–187].
3.3 Dynamic structure factor
The Bragg spectra provide access to the dynamic structure factor, S(q, ω), which
characterises the cross section for inelastic scattering. The momentum and energy
transferred to the cloud lifts the ground state, |0⟩, into an excited state, |n⟩, accord-ing to equation 3.2, and the particle gains a momentum q. As mentioned above,
this is a stimulated process. The many-body system is perturbed by the periodic
potential, Vr, formed by the electric fields of the Bragg laser beams. The Fourier
transform of the perturbing potential, Vq = 2~ΩR, can be expressed in terms of
the two-photon Rabi frequency, ΩR, which expresses the coupling strength between
the atom and light field. Implicitly, it is assumed that at t = −∞ the system is
governed by the unperturbed Hamiltonian. The interaction between the atoms and
the probe, Hint = HBr, may be written as
Hint =Vq2
(δρ†qe
−iωt + δρqeiωt). (3.3)
The Fourier transform of the particle density (operator) is defined as
ρq =
∫d3rρ (r) e−iqr =
∑j
∫d3rδ (r− rj) e
−iqr =∑j
e−iqrj . (3.4)
The fluctuations generated by the external field of the particle density, δρq = ρq−ρ0,oscillate about the average value, ρ0. Let Hint couple weakly to the ground state of
the fermionic cloud |0⟩. This implies that the eigenstates are unperturbed before,
52 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
|0,p⟩, and after, ⟨n,p− q|, the scattering event. Hence,
⟨n,p− q |Hint| 0,p⟩ = Vq⟨n∣∣δρ†q∣∣ 0⟩ ≡ Vq
(δρ†q)n0. (3.5)
The density fluctuations, δρ†q, depend on the matrix elements between the exact
many-particle eigenstates in the absence of the Bragg pulse. The external particle
thus probes the density fluctuations in the system. The probability, W (q, ω), of
momentum and energy transfer per unit time is readily given by the golden rule of
second order perturbation theory, i.e. the summation runs over all states, |n⟩, thatare coupled to the ground state, |0⟩, by the density fluctuations δρ†q, so
W (q, ω) = 2π |Vq|2 S (q, ω) (3.6)
with
S (q, ω) = ~∑n
∣∣∣(δρ†q)n0∣∣∣2 δ (ω − ωn0) (3.7)
and ωn0 = ωn − ω0. ω and q depend on each other as in equation 3.2. Thus
S(q, ω) embodies all the properties of the many-body system that are relevant to
the scattering of the Bragg photon and it represents the maximum information
of the system behaviour that can be obtained in a particle scattering experiment.
The distribution of squared matrix elements for each allowed transition reflects the
excitation spectrum of the system. At T = 0, S (q, ω) is real and positive, i.e.
S (q, ω ≤ 0) = 0, since all excitation frequencies, ωn0, are necessarily positive. The
spectral density of the state δρ+q |0⟩ corresponds to correlations between density
fluctuations at different times. The correlation between a density fluctuation at
time t = 0 and at time τ is expressed by the correlation function, S (q, τ), which is
the Fourier transform of S (q, ω) with respect to time. In the high momentum limit
collective features are irrelevant, since the probe can directly scatter out individual
constituents, but the momentum distribution n (p) of the sample can be investigated.
In Fermi gases the momentum distribution is a key quantity as it dominates the
dynamics and properties. In the high momentum limit, S(q, ω) can be written as
SIA (q, ω) =
∫dpn (p) δ
(~ω − (p+ q)2
2m+
p2
2m
). (3.8)
This impulse approximation is uniquely determined by the momentum distribution
of the cloud [188]. From here, one can see that a resonance occurs at the recoil
3.3 DYNAMIC STRUCTURE FACTOR 53
frequency, ωR = ~q2/2m, when p = 0. Note that for a noninteracting Fermi gas
the spectral function of S(q, ω) reflects the particle density distribution, n0(p), of
an ideal Fermi gas [80], see figure 3.2.
0 0.5 1 1.5 20
2
4
6
8
10
ω/ωR
S0 (k,ω
)
Figure 3.2: The dynamic structure factor for an ideal Fermi gas, S0IA(k, ω), in the
impulse approximation at T = 0 shows the momentum transferred with a Braggpulse. The resonance occurs at ω = ωR.
For a non-interacting Fermi gas, an illustration is given by figure 3.3a. The
ground state, |0⟩, consists of all plane waves filled up to the Fermi sphere with
radius pF , or ~kF . When the operator δρ+q acts on the ground state, single-particle
transitions are induced. A particle in state p is annihilated, cp, and in state p + q
created, c†p+q, such that δρ†q =∑
p c†p+qcp for the wave vector p and spin σ, which is
implicitly included in the summation. ρ+q represents a superposition of particle-hole
pairs each of net momentum q since particle scattering is described equally well as
creation of a particle-hole pair. Because of the exclusion principle, a given pair in
the ground state can only be excited if the state p is filled and the state p + q is
empty. Hence, the allowed values of p are located at the intersection between the
ground state Fermi sphere with pF and the sphere with p′F of empty states which
is displaced by an amount −q. If k < 2kF , the Pauli exclusion principle begins to
reduce the number of available final states. The excitation spectrum of particle-hole
54 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
(a) (b)
Figure 3.3: (a) Left: particle with Fermi surface at radius pF . Right: the particlegains momentum -q. The fermionic character restricts scattering to particles whichfulfil initial momentum states with p < pF as well as final states with |p+ q| > p′F .All of these excitations are represented by the grey shaded area. The closer therelative momentum transfer, (q/pF ), is to unity, the more excitations can contributeas the probing wavelength becomes closer to the correlation length. In contrast, alarge momentum transfer, q/pF ≫ 1 corresponds to excitations in the corners ofthe grey shaded area and the signal of correlated pairs is smaller. (b) Visualisationof particle-hole excitations for a non-interacting Fermi gas in time-of-flight. Atomsare excited when they are resonant with the Bragg momentum. The out-scatteredatoms travel within a release time and leave a slice in the momentum distributionof the parent cloud [189].
pairs with total momentum q will form a continuum limited by
− qpFm
+q2
2m≤ ω0
pq ≤ qpFm
+q2
2mfor k > kF . (3.9)
At T = 0, the dynamic structure factor in the ideal Fermi case, S0 (q, ω), vanishes
outside this finite range, see figure 3.2. This feature changes for real Fermi liquids.
For all values of q, S0 (q, ω) → 0 when ω → 0 due to the proportionality to ω
(which is at the margins of the intersecting surfaces where the shifted Fermi sphere
is not very different from the Fermi sphere of the ground state). Such behaviour is
a direct consequence of the exclusion principle and of the corresponding scarcity of
low-energy excitations as the number of available empty states is much smaller.
3.4 STATIC STRUCTURE FACTOR 55
3.4 Static structure factor
With the simplification that the momentum transfer is essentially independent of
the energy transfer, the probability to transfer energy and momentum to a particle,
given by equation 3.6, can be integrated over all permissible frequencies∫ ∞
0
dωW (q, ω) = 2π |Vq|2NSq. (3.10)
The definition of the static structure factor follows as
Sq = S (q) =~N
∫ ∞
0
dωS (q, ω) . (3.11)
NSq is the non-energy-weighted moment and is governed by the momentum of
the scattered particles. It is equal to the Fourier transform of S (q, τ) taken for
t = 0, so NSq = ⟨0| ρ†qρq |0⟩. The static structure factor provides a measure of the
instantaneous density correlations in the system or mean square density fluctuations
at momentum q and provides an integrated response of the equilibrium configuration
of S (q, ω). If∣∣∣ψ⟩ is the field operator, then S(k) is related to the two-body density
matrix
n(2) (r1, r2) =⟨ψ† (r1) ψ
† (r2) ψ (r1) ψ (r2)⟩
(3.12)
by
S (q) = 1 +1
N
∫dr1dr2
[n(2) (r1, r2)− n (r1)n (r2)
]eiq·(r1−r2). (3.13)
As only the relative distance, s = r1 − r2, is relevant in uniform systems
S (q) = 1 + n
∫ds [g (s)− 1] eiq·(s). (3.14)
The pair correlation function, g (s) = n(2) (s) /n2 is normalised to n =⟨ψ†ψ
⟩and
is given through the Fourier transformation of the static structure factor
g (s) = 1 +1
n (2π)3
∫dq [S (q)− 1] e−iq·(s). (3.15)
At high momenta, S (q → ∞) approaches unity because momentum states are un-
correlated such that g(s) → 1 and incoherent processes dominate. Mathematically
speaking, only the diagonal elements in ρ†qρq =∑
lj exp [iq (rl − rj)] make the sig-
56 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
nificant contributions.
3.5 f-sum rule
Sum rules ensure that the many-particle systems converge in certain limits. In this
context, particle conservation relates the scattered particles with the involved en-
ergy states through the f-sum rule. The derivation is outlined below. The strongly
interacting Fermi gas can be regarded as an incompressible liquid as in the hydro-
dynamic picture. There, the fundamental equation of continuity guarantees particle
conservation. In the Heisenberg picture this is
i~dρqdt
= [ρq, H] . (3.16)
The particle density, ρq, is described by equation 3.4 and has no time dependence.
One needs to calculate [ρq, H] with the Hamiltonian of the system
H =∑j
[p2j
2m+ U (rj)
]+
1
2
∑j =l
Vr (rj − rl) , (3.17)
which has an external potential U (rj) and an interparticle potential Vr (rj − rl).
Both potentials are independent of the velocity. As the particle density, ρq, is also
only dependent on the position, it commutes with the potentials and the only con-
tribution from [ρq, H] stems from the kinetic energy term. In a detailed calculation
one can use standard commutator relations to show that [ρq, H] is equal to the
scalar product, ~(jq · q), of the current density fluctuation, jq, and the wave vector,
q. Its eigenvalues, ~ωn0 (ρq)n0 = ~(jq · q)n0, lead to the matrix elements between
the eigenstates, ~ωn0 (ρq)n0, of the real system, which are the transitions from the
ground state. As an interim result, particle conservation is linked to the permissible
energy states, n, in the scattering event by
i~dρqdt
= [ρq, H] = ~ωn0 (ρq)n0 . (3.18)
To calculate the energy-weighted momentum of S(q, ω), the double commutator is
evaluated for the ground state,⟨0∣∣[[ρq, H] , ρ†q
]∣∣ 0⟩. On the one hand, one obtains
with equation 3.18
⟨0∣∣[[ρq, H] , ρ†q
]∣∣ 0⟩ = 2~∑n
ωn0
∣∣∣(ρ†q)n0∣∣∣2 . (3.19)
3.6 LINEAR RESPONSE FUNCTION 57
On the other, one can explicitly calculate the double commutator for the kinetic
part of the Hamiltonian, equation 3.17, which may be written for N particles as
[[ρq, H] , ρ†q
]=Nq2~2
m. (3.20)
The double commutator leads to
~∑n
ωn0
∣∣∣(ρ†q)n0∣∣∣2 = Nq2~2
2m. (3.21)
Hence, the f-sum rule follows with equation 3.7 as
~2∫ ∞
0
dωωS (q, ω) =Nq2~2
2m. (3.22)
It bears this name because the oscillator strength for each transition is defined as
f0n = (2m/q2)∑
n ωn0
∣∣∣(ρ†q)n0∣∣∣2 such that∑f0n = N , e.g. [4]. The strength of the
f-sum rule lies in its validity for a wide range of interacting many-body systems,
independent of statistics and temperature, and it is well-known in atomic physics.
In the long wavelength limit, q → 0, other sum rules also exist, for instance, the
compressibility sum rule, which can be used to obtain the speed of sound.
3.6 Linear response function
In this work, the true linear response function is not explicitly calculated but its
imaginary part is measured and linked to S(q, ω). For these experiments, the linear
response is restricted to the density-density correlations. The idea is, as stated
above, that the weak perturbation of the Bragg photons on the Fermi gas results
in density fluctuations which are proportional to the linear response function and
the general formalism is provided by first order perturbation theory. Amongst the
phenomena of non-equilibrium dynamics, linear response theory is limited to small
disturbances from the equilibrium state where the Born approximation is valid, that
is, the many-body system is assumed to be undepleted by the scattering event. This
is a widely applicable concept in natural systems. The standard linear response
function, χ (r, t), is defined by
⟨ρ (r, t)⟩ =∫d3r′
∫dt′χ (r− r′, t− t′)ϕ (r′, t′) , (3.23)
58 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
where a fluctuation of the physical quantity ⟨ρ (r, t)⟩ is caused by an external po-
tential ϕ (r′, t′). In Fourier space the expression is more compact through the re-
placement of space and time with momentum and frequency. The scalar potential
is real, ϕ (q, ω) = ϕ∗ (−q,−ω), and is part of the time dependent Hamiltonian. For
the solution the Dirac picture is useful where the unperturbed part is given in the
Schrodinger picture and the external perturbation, Hint, follows the Heisenberg pic-
ture with time-dependent operators. All the transitions, ωn0, from the ground state,
|0⟩, to any excited state, |n⟩, are coupled by the density fluctuations. For Bragg
scattering it follows that ϕ (q, ω) → Vq and Hint → HBr. In experiments, the atoms
in the unperturbed system (t = −∞) respond to the Bragg pulse, whose rising slope
has a finite width. For the calculation of the response function, one needs adiabatic
boundary conditions and causality, because a physical response to a perturbation
happens only after the perturbation and the evolution of the system should not di-
verge after the perturbation. These conditions are governed by introducing a small
parameter η → 0. Independent of the details of the scattering potential and using
again equation 3.7, the connection between the linear response and S(q, ω) is defined
as
χ (q, ω) =
∫ ∞
0
dω′S (q, ω′)
1
ω − ω′ + iη− 1
ω + ω′ + iη
. (3.24)
The dynamic structure factor contains the spectral information on the density-
density response function. The result gives the linear response function in terms
of the true eigenstates of the system, ω. The symmetric notation is necessary to
account for reversible excitations, where S (q, ω) is already symmetric. In the high-
frequency limit equation 3.24 becomes
χ (q, ω) → 2
ω2
∫ ∞
0
dω′S (q, ω′) =Nq2
mω2(ω → ∞) (3.25)
and this asymptotic form of χ(q, ω) is unaffected by the interaction between the
system particles. The real and imaginary parts of the response function χ = χ′+ iχ′′
are
χ′ (q, ω) =
∫ ∞
0
dω′S (q, ω′)P
(2ω′
ω2 − ω′2
)(3.26)
χ′′ (q, ω) = −π [S (q, ω)− S (−q,−ω)] , (3.27)
3.7 RESPONSE AT FINITE TEMPERATURE 59
where P is called the principal part, which is a mathematical residue from the Dirac
relation when integrating in the limit of η → 0. These equations are linked through
the well-known Kramers-Kronig relations, [190,191]. S(q, ω) provides a measure of
the real transitions induced by the probe, i.e. of the energy transfer from the probe
to the system. The imaginary part measures the inelastic cross section in the dis-
sipative scattering event. χ(q, ω) does not provide any more physical information
than S(q, ω). At T = 0, the quantity which represents the physical information
in the most compact form is S(q, ω); it gives the density fluctuation excitation
spectrum as a positive definite function. But at finite temperatures the relation
between S(q, ω) and χ′′(q, ω) is more complicated. Bragg spectra have been numer-
ically calculated based on the linear response function, χ(q, ω) [192]. In general,
the formalism of the response function is applicable to any quantum liquid, whether
Bose or Fermi, charged or neutral, normal or superfluid, and is a compact way to
describe experimental measurements involving weakly coupled external probes. In
the hydrodynamic regime, such as for a strongly interacting Fermi gas, the corre-
lation functions need to be determined by macroscopic considerations, since details
of the single particle excitation spectrum have been washed out by the collisions.
Section 3.8 will focus on this.
3.7 Response at finite temperature
At finite temperatures, T , a statistical description is necessary to account for the
thermal equilibrium of the initial (t = −∞) configuration. In the canonical ensemble
of statistical mechanics, the exact eigenstates |m⟩ of the system Hamiltonian possess
an energy Em and the probability of finding the system in state |m⟩ is given by
W (m) =e− Em
kBT
Z(3.28)
where the partition function is Z =∑
m e−Em/(kBT ), i.e., in thermal equilibrium
the probability of finding a particle in either of the two spin states is related to the
Boltzmann factor. A consequence of the interaction between the Bragg pulse (a time-
dependent external probe) and the fermionic many-particle system is the possibility
of energy transfer to and from the probe. The principle of detailed balancing is used
to link S (q, ω) with the imaginary part of the density-density response function
χ′′ (q, ω), so equation 3.27 is modified [76]. Unlike the general expressions in the
previous section, at finite temperatures they are different as S(q, ω) is a direct
measure of the fluctuations in the system, while χ′′ (q, ω) describes the dissipative
60 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
part of the system response. The integrated form of detailed balancing is the well-
known fluctuation-dissipation theorem [193,194].
3.7.1 Consequences for the dynamic structure factor and
the response
Within the Born approximation the dependence on the statistical weight, equa-
tion 3.28, enters the probability per unit time that a probe particle transfers mo-
mentum q and energy ω to the system as
W (q, ω) = 2π |Vq|2 Z−1∑mn
e− Em
kBT
∣∣∣(δρ†q)nm∣∣∣2 δ (ω − ωnm) . (3.29)
The result follows from second order perturbation theory for a transition from a
given state |m⟩ to another state |n⟩. ~ωnm is the exact energy difference, En −Em.
For the density-density response function one finds
χ (q, ω) = Z−1∑mn
e− Em
kBT
[∣∣∣(δρ†q)nm∣∣∣2 1
ω − ωnm + iη− 1
ω + ωnm + iη
](3.30)
η→0= Z−1
∑mn
e− Em
kBT
∣∣∣(δρ†q)nm∣∣∣2 2ωnm
ω2 − ω2nm
.
which was first stated by Kubo [195] for any linear operator. The spectral excitations
are now thermally weighted, and therefore S(q, ω) must depend on the temperature,
since it contains the properties of the many-particle system
S (q, ω) = Z−1∑mn
e− Em
kBT
∣∣∣(ρ†q)nm∣∣∣2 δ (ω − ωnm) . (3.31)
Equation 3.7 is recovered for T = 0, where S(q, ω) identically vanishes for ω < 0.
The excitation energies ωn0 are always positive for a system in the ground state.
3.7.2 Detailed balancing
In general, if the system can undergo a transition from a state m with equilibrium
probability πm to a state n with equilibrium probability πn, the transition probability
is Wmn and its reciprocal transition is Wnm. Detailed balancing
πmWmn = πnWnm, (3.32)
3.7 RESPONSE AT FINITE TEMPERATURE 61
states that in thermal equilibrium all microscopic processes are compensated for by
their reciprocal processes. In this context, the equilibrium probabilities are given by
the Boltzmann factors πm = exp(−~ωm/(kBT )) and πn = exp(−~ωn/(kBT )) at the
thermal equilibrium temperature, T , and ~ωmn = Em −En. Recalling equation 3.29
and that the probe can also absorb momentum −q and energy −ωnm = +ωmn from
the system, the probability per unit time is written as
W (−q,−ω) = 2π |Vq|2 Z−1∑mn
e− Em
kBT
∣∣∣(ρ†q)nm∣∣∣2 δ (ω + ωnm) (3.33)
= 2π |Vq|2 S (−q,−ω) .
and S(q, ω) is proportional to the transfer probability. In that case, the principle of
detailed balancing states that
W (q, ω)
W (−q,−ω)=
S (q, ω)
S (−q,−ω)= e
~ωkBT (3.34)
which is satisfied by equation 3.29 and equation 3.31. This can also be seen when
interchanging indices in equation 3.31
S (q, ω) = Z−1∑mn
e− Em
kBT
∣∣∣(ρ†q)nm∣∣∣2 δ (ω − ωnm) (3.35)
= Z−1∑nm
e− En
kBT
∣∣∣(ρ†q)mn
∣∣∣2 δ (ω + ωnm)
= Z−1∑nm
e−En−Em
kBT e− Em
kBT∣∣(ρq)nm∣∣2 δ (ω + ωnm)
= e~ω
kBT S (−q,−ω) .
Notice that if the system is in thermal equilibrium, and hence equation 3.34 holds,
the total energy rate is always positive due to causality.
3.7.3 Fluctuation-dissipation theorem
The dynamic structure factor is related to the density response function through its
imaginary (dissipative) part which specifies directly the steady state energy transfer
from an oscillating probe (Bragg lattice) to the many-particle system (strongly in-
teracting Fermi gas). At finite temperature, the imaginary part, equation 3.27, may
be written explicitly as
χ′′ (q, ω) = −π[1− e
− ~ωkBT
]S (q, ω) (3.36)
62 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
which is known as the fluctuation-dissipation theorem [193, 194]. It is one of the
most fundamental results of quantum statistics. The derivation is very arduous.
In principle, a system in thermal equilibrium is considered, for which the theorem
states that the reaction to a small external perturbation is the same as the reaction
to spontaneous fluctuations and that the dissipative part is proportional to these
fluctuations. With regard to S(q, ω), one can find an explicit relation between the
microscopic dynamics and macroscopic reactions to time-dependent perturbations.
These dynamics are accessible with Bragg spectroscopy. In contrast, the dissipation-
fluctuation theorem allows one to use microscopic models in a statistical equilibrium
for predictions of macroscopic properties which might deviate from equilibrium.
At finite temperatures, χ′′ (q, ω) and S (q, ω) have a slightly different temperature
dependence. The dynamic structure factor can be presented in a more symmetric
way
S (q, ω) =S (q, ω) + S (q,−ω)
2= S (q, ω)
1 + e− ~ω
kBT
2. (3.37)
The dissipative part is introduced by equation 3.36 such that
S (q, ω) = − 1
2πχ′′ (q, ω) coth
(1
2
~ωkBT
). (3.38)
The mean square thermal fluctuations can be written as
⟨ρ†qρq
⟩= NSq = −~
∫ ∞
−∞
dω
2πχ′′ (q, ω) coth
(1
2
~ωkBT
). (3.39)
This integrated form of the detailed balance is a form of the fluctuation-dissipation
theorem. The sum rules for χ′′ (q, ω) and S(q, ω), which were derived in the zero
temperature case, are also applicable at finite temperatures. The f-sum rule can be
rewritten as
2~2∫ ∞
−∞dωωS (q, ω) = −~2
π
∫ ∞
−∞dωωχ′′ (q, ω) =
Nq2~2
m. (3.40)
The above considerations are particularly important at finite temperatures. In fact,
at T = 0, χ′′(q, ω) and S(q, ω) coincide for positive ω and only processes transferring
energy to the system are allowed, while at finite temperatures the two functions differ
significantly if ~ω/(kBT ) is small. All temperature dependent Bragg spectra will be
represented in terms of S(q, ω) in the later chapters since the additional factor of
(1+exp [−~ω/(kBT )])/2 is very small in the applied temperature range and S(q, ω)
3.8 DYNAMIC STRUCTURE FACTOR FROM BRAGG SPECTRA 63
is conventionally used in inelastic scattering experiments.
3.8 Dynamic structure factor from Bragg spectra
Bragg spectroscopy is a powerful tool to investigate the dynamics in ultracold gases.
This section shows how the microscopic details of strongly interacting Fermi gases
given by S(q, ω) can be measured under macroscopic considerations of an atomic
distribution. All results for this thesis are extracted from absorption images. This
technique is well understood and has been applied since the beginning of ultracold
atomic gas experiments. Despite the convenience of gaussian fitting, which is often
applied when the signal-to-noise drops, more careful methods are necessary to adapt
to accurately fit the density profiles of Fermi clouds to determine their properties. A
series of absorption images (time-of flight technique) at different Bragg frequencies
are recorded. The line profiles of each image are the basis to generate an energy
resolved Bragg spectrum. As the sample is excited by a momentum kick, the re-
sponse of the trapped cloud is manifested in a distortion of the cloud shape and the
response is proportional to the number of scattered particles, Nsc. There are two
possibilities to evaluate the response. One way is to look at the distribution shortly
after the Bragg pulse. The integrated line profile is then asymmetric. Traditionally,
the centre of mass displacement [78] or the number of optically excited atoms is con-
sidered [26] and more recently the counted photons in the Bragg beams [196]. The
centre of mass displacement reflects the first moment of the line profile. However,
also the second moment of the line profile contains valuable information. Another
way is to apply a long hold time after the Bragg pulse, so that the scattered atoms in
the trap reach energetic equilibrium. Then, the line profiles are symmetric and the
information is linked to the change in the width of the cloud. The evaluation of the
width, which is associated with the transferred energy, has been used in experiments
with bosons in optical lattices [185–187]. The details on the acquisition of Bragg
spectra are described later as every experiment has an individually modified image
process. In the following sections, it is important to make the connection between a
spectrum and S(q, ω), and concentrate on the application of interdisciplinary sum-
rules helping to distil physical information stored in a Bragg spectrum.
Note that atoms can scatter by absorbing a photon from either of the two
laser beams, and that the response of the system measures the difference rather
than S(q, ω) itself, as seen in equations 3.27 and 3.36
χ′′ (q, ω) = π [S (q, ω)− S (−q,−ω)] = πS (q, ω)[1− e
− ~ωkBT
](3.41)
64 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
This is an important difference with respect to other scattering experiments (like
neutron scattering from helium) where, by detecting the scattered probe, one mea-
sures directly S(q, ω).
3.8.1 Measuring the dynamic structure factor
The principle of Bragg scattering was described in section 3.2.1. The evolution of
the system is governed by the time dependent Schrodinger equation
i~∂
∂tϕ (r, t) = Htotϕ (r, t) =
[− ~2
2m∇2 + Vtrap (r) + Θ (t)Hint
]ϕ (r, t) . (3.42)
The unperturbed system has the kinetic term and the harmonic confinement Vtrap (r)
being produced by the trap potential. Θ (t) is the Heaviside step function to switch
on the perturbationHint given by equation 3.3. Using first order perturbation theory,
one can calculate the centre of mass displacement in the direction of the Bragg pulse
z, as
i~dPz
dt= ⟨[Pz, HBr]⟩ . (3.43)
The overall trap frequency ν and width d ⟨σ2⟩ enter the expression for the energy
dE = (1/2)mν2d ⟨σ2⟩, which can be obtained from
i~dE
dt= ⟨[H,HBr]⟩ . (3.44)
The centre of mass displacement and the width of the atomic distribution are linked
to the response by
1
q
dPz
dt=
1
ω
dE
dt= 2
(Vq2
)2
S (q, ω)(1− e−~ω/(kBT )
). (3.45)
Note that Vq ≡ 2~ΩR.
3.8.2 Dynamic structure factor from the centre of mass dis-
placement
The net momentum imparted to the cloud can be observed through the centre of
mass displacement after the cloud is released from the trap, independent of the cloud
shape. The time derivative of the momentum is given by the Heisenberg picture, see
equation 3.43. This calculation is reported elsewhere [80, 197]. One can rewrite the
momentum as velocity of the centre of mass displacement Pz (t) = m(dX (q, ω) /dt).
3.8 DYNAMIC STRUCTURE FACTOR FROM BRAGG SPECTRA 65
For a given Bragg pulse, τBr, and time of flight τTOF , the centre of mass displacement
travels ∆X(q, ω) = Xfin(q, ω) − X0, where X0 denotes the cloud position at rest
and Xfin(q, ω) the final centre of mass of the cloud after the Bragg pulse and time-
of-flight. Including detailed balancing, see equation 3.34, an integration shows
∆X (q, ω′) = (3.46)
τTOF
m
2q
~
(Vq2
)2 ∫dω′S (q, ω′)
(1− e−~ω′/(kBT )
) 1− cos [(ω − ω′) τBr]
(ω − ω′)2.
The fraction indicates the signal is convoluted with the spectral components of the
Bragg pulse. For all permissible excitation frequencies ω′, the integration over the
fluctuation frequency is constant. It follows with the definition of the static structure
factor, equation 3.11,∫dω
∆X (q, ω)
1− e−~ω/(kBT )∝∫dωS (q, ω) = NS (q) . (3.47)
The centre of mass displacement leads via the static structure factor to the pair
correlation function, see equation 3.14. The transferred momentum reflects in the
number of scattered particles (atoms or pairs), Nsc, such that
∆P (q, ω) = md∆X (q, ω)
dt= Nsc · q = Nsc · ~k. (3.48)
According to equation 3.47, the transferred momentum is therefore proportional to
the dynamic structure factor, ∆P (q, ω) = Nsc~k ∝ S (q, ω).
3.8.3 Dynamic structure factor from the width
Starting from equation 3.44, the Bragg Hamiltonian depends on the local density
and only the kinetic energy term in H contributes to the above commutator. The
calculation is similar to the derivation of the f-sum rule [197]. One finds
∆E (q, ω′) = (3.49)
τTOF2
π
(Vq2
)2 ∫dω′ω′S (q, ω′)
(1− e−~ω′/(kBT )
) 1− cos [(ω − ω′) τBr]
(ω − ω′)2
which holds for t > 0. The transferred energy from Bragg pulse is accessible via the
width of the cloud distribution, ∆E = (1/2)mν2∆ ⟨σ2⟩. As the trap frequency, ν, is
fixed, one needs to measure only the increase in width, ∆ ⟨σ2⟩ =⟨σ2fin
⟩−⟨σ2
0⟩, where⟨σ2fin
⟩is the final width after the Bragg pulse and ⟨σ2
0⟩ the width of the unperturbed
66 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
cloud. The static structure factor follows again from equation 3.11∫dω
∆σ2 (q, ω)
1− e−~ω/(kBT )∝∫dωωS (q, ω) . (3.50)
The energy transfer, ∆ ⟨σ2 (q, ω)⟩, depends on the product of the Bragg frequency
(energy) and the number of scattered particles ω ·Nsc, such that
∆E (q, ω) =1
2mν2∆
⟨σ2 (q, ω)
⟩= ~ω ·Nsc (3.51)
With equation 3.50, one can see that the transferred energy in terms of the width is
proportional to first moment of the dynamic structure factor, ∆E(q, ω) = ~ω ·Nsc ∝∆ ⟨σ2(q, ω)⟩ ∝ ωS(q, ω).
3.8.4 Application of sum-rules
Sum rules provide a model-independent tool to determine the moments of the dy-
namic structure factor avoiding the explicit determination of the response function
and the requirement of the full solution of the Schrodinger equation. In general, the
moments of order p are calculated as
mp = ~p+1
∫ +∞
−∞dωωpS (k, ω) (3.52)
Equation 3.11 states that the static structure factor is the zero order moment
NS (k) = ~∫ +∞
−∞S (k, ω) dω (3.53)
and thus proportional to the atom number Nsc [76]. The dynamic structure factor
in first moment is also called the f-sum rule
NER = ~2∫ +∞
−∞ωS (k, ω) dω (3.54)
for the known recoil energy ER = ~k2/2m. As both of the equations involve Nsc,
the normalisation of the static structure factor yields
S (k) =ER
~
∫ +∞−∞ S (k, ω) dω∫ +∞−∞ ωS (k, ω) dω
. (3.55)
3.9 SUMMARY 67
The proportionality constants in equation 3.47 and 3.50 depend on experimental pa-
rameters and spectral information of the Bragg pulse such as the Rabi-cycle (Bragg
laser detuning, pulse duration, Bragg laser power), time-of-flight, laser line widths,
scattering length at different interaction strength 1/(kFa) etc.. Equation 3.55 plays
the central role of extracting information from the spectra and allows one to obtain
an absolute number for S(k) without any free scaling parameter since both inte-
grals have the same proportionality constant, which therefore cancel. The method
of centre of mass displacement, equation 3.47, was used in the very first spectra [80]
to demonstrate the decrease of the static structure factor from 2 → 1 as the relative
momentum, k/kF = 5, was transferred at different interaction strengths across the
Feshbach resonance ranging from the BEC limit to the BCS limit, see figure 6.7a.
Equations 3.53 to 3.55 can be applied to compare with a now available theoretical
prediction [82]. Compared to the previously reported result [80], the accuracy of the
data points is much better than the earlier results, which relied on an experimental
determination of the two-photon Rabi cycling period ΩR.
3.9 Summary
Inelastic two-photon Bragg scattering with photons of large momenta can be used
to probe the momentum distribution of strongly interacting Fermi gases. Linear
response theory provides a model-independent way to describe the spectral func-
tions in many-body systems obtained from Bragg spectroscopy. The Bragg photons
couple weakly to the fermionic cloud and quantities are derived within first or-
der perturbation theory. The spectral function is stored in the dynamic structure
factor, S(q, ω), which only depends on the momentum and energy transfer of the
photon. The net momentum transfer is given by the static structure factor, S(q),
after energy integration of the dynamic structure factor. The static structure factor
measures instantaneous density fluctuations and is related to the two-body correla-
tion function. Within linear response theory, sum rules are found in several limits,
for instance the f-sum rule. The fluctuation-dissipation theorem and detailed bal-
ancing are equivalent formulations to include finite temperature in the dynamic and
static structure factors. This theoretical background is applied to the measurement
of pair-correlations in Fermi gases. The interaction potential in the Hamiltonian is
given by the periodic potential of the Bragg beams. Measurements of the dynamic
structure factor can be extracted from the centre of mass displacement or energy
of atomic distributions perturbed by the Bragg pulse. The degree of two-body cor-
relations in the Fermi gas is reflected in the static structure factor which can be
68 CHAPTER 3. STRUCTURE FACTOR OF A FERMI GAS
normalised with the f-sum thus rule avoiding a dependence on the atom number
and other experimental parameters. Details on the acquisition of Bragg spectra are
presented in chapters 6 and 7 together with image evaluation techniques.
Chapter 4
Degenerate quantum gas machine
Since the first production of bosonic [9–11], ultracold degenerate gases the experi-
mental setups have undergone vast technical developments to cover the possibilities
for investigating cold atoms. Our apparatus is designed to trap and evaporatively
cool fermionic 6Li atoms and study them in a wide range of magnetic fields.
4.1 Introduction
The experiments in this thesis have been carried out in an ultra high vacuum glass
cell. The original setup was designed and built by Fuchs [198] with the main goal
to produce molecular BECs [199]. All the details concerning the chamber design,
vacuum system and pumping can be found in that thesis. Major changes in the laser
system for cooling and repumping as well as the installation of the offset locking for
imaging were made before the first Bragg spectroscopy experiments [189]. The most
recent detailed description of the apparatus, especially details on the Feshbach coils
and the computer control facilities, is given in [200]. A brief outline of the principal
stages of the experimental procedure before evaporation to a quantum superfluid is
outlined in the following. Details on the principles can be found in textbooks such
as [114, 201, 202]. The Feshbach coils are essential for these experiments and there-
fore recalled in section 4.2. The setup for the two optical dipole traps is explained
in section 4.3 with regard to the characteristic parameters for chapters 6 and 7. The
Bragg beams represent the spectroscopic tools for probing the cloud and are the
subject of section 4.4. Detection with absorption imaging is an important require-
ment to obtain qualitative information on the atomic distributions. The central
goals of the imaging setup are to minimise noise, obtain the correct atom numbers
and correct cloud shape, which is of special interest for the determination of the
temperature, as described in chapter 5.
69
70 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
Figure 4.1: Schematic setup of the apparatus in top and side view [189]. The mainparts are the lithium oven, Zeeman slower and glass cell with surrounding coils. Thepressure changes between the left and right pumping chamber.
Precooling atoms
To generate an ultracold Fermi gas the atoms have to undergo several stages of
cooling. As the experiments take place in a thermally isolated high vacuum envi-
ronment, cooling means extracting kinetic energy. The source of 6Li atoms is stored
in solid form in an oven that heats the lithium above the sublimation temperature
to a vapour with a thermal Boltzmann distribution released from an effusive source.
The vacuum system without the laser system is sketched in figure 4.1 [189]. The
pressure changes from 1.2 · 10−4 mbar in the lithium oven (T = 450C) to ∼ 1.0 ·10−11 mbar in the glass cell. The water-cooled Zeeman slower coil is wound on a 30
cm-long stainless steel tube with increasing inner diameter from one end to the other.
It is directly connected to a custom-made1 glass cell made from Vycor quartz2. The
cell dimensions are 12 × 3 × 3 cm3. On the outside surface an antireflection coating
is chosen for 1030 nm as well as 670 nm light to optimise the transmission for the
cooling, imaging as well as trapping light. The magnetic trapping potential is gen-
erated by a set of water-cooled coils in anti-Helmhotz configuration. The symmetry
1Hellma2Base Vycor 7913
4.1 INTRODUCTION 71
axis of the magnetic trap is in the vertical direction and defines the z-axis of the
experimental setup. Three extra sets of compensation coils provide separate offset
fields to shift the MOT into the optimum transfer position.
Atoms below a certain cut-off velocity are captured by the Zeeman laser beam
and the magnetic field of the Zeeman coils. While the atoms travel against the Zee-
man beam in the coil, they are slowed down due to a directional process of multiple
photon absorption and emission. At the end of the tube, their final velocity is small
enough that they do not overshoot the potential created by the magneto-optical trap
(MOT). There, the atoms are pushed to the trap centre by the cooling and confining
MOT forces. The MOT is produced by a combination of six counter-propagating
circularly polarised laser beams (MOT beams) that intersect in the zero magnetic
field of two magnetic coils in an anti-Helmholtz configuration. Basically, the MOT
beams maintain the cooling process in momentum space by photon absorption-
emission cycles while the magnetic field gradient provides a spatial dependence on
the force which pushes the scattering atoms back to the trap centre. The atomic
temperature is limited to about 200 µK as the fluorescent scattering mechanism
imparts a recoil momentum to the atoms. To overcome three orders of magnitude
in temperature to enter the degeneracy regime, the MOT light is switched off and
further cooling is performed after compression and transfer into a far-detuned single
optical dipole trap. Here, evaporative cooling is performed and the lowest tempera-
tures are achieved down to 0.1 T/TF , where TF is the Fermi temperature. Once an
ultracold gas in the superfluid regime is created the experiments can start. The de-
tection is performed with destructive absorption imaging, where the shadow picture
is taken on a CCD chip for further evaluation processes.
The delicate technique of loading and evaporating atoms demands a precise
control of the timing and triggers of the voltage-regulated experimental control pa-
rameters, such as used for the laser powers and frequencies, magnetic fields and
camera trigger. Three National Instruments PCI output boards3 provide 24 analog
output channels with output voltages ranging from − 10 V to + 10 V. The control
program and graphical user interface is written in LabView4.
The laser system with 671 nm light incorporates the locking of lasers to the
lithium absorption lines, the cooling system for the atoms, the Bragg lasers as an
independent manipulation tool and the imaging laser for detection. The frequency
control in the laser setup is referenced to a 6Li D2 sub-Doppler resonance obtained
from lithium vapour in saturation spectroscopy (22S1/2 → 22P3/2 transition). Once
3National Instruments: PCI 6733 and PCI 67134National Instruments: LabView 6
72 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
Figure 4.2: Laser locking frequencies in the 6Li energy level diagram.
4.2 FESHBACH COILS 73
the reference frequency is locked all other frequencies, for example for the slowing,
cooling and imaging, can be achieved with acousto-optic modulators (AOMs). The
relevant energy levels and the locking scheme are summarised in figure 4.2.
Figure 4.3 shows the optical setup for cooling. An external cavity diode laser5
(ECDL) with an output power of 15 mW provides the light for the spectroscopy.
The pump beam is frequency-shifted with an AOM by 208 MHz before entering the
spectroscopy loop. While the crossover peak is pumped with a 44 MHz detuning
(velocity selective pumping), the probe beam monitors the transmitted intensity on
a photo diode (PD). On this absorption dip the lock-in amplifier technique with a
250 kHz external modulation can be used to obtain an error signal which in turn
provides the zero-crossing slope for the electronic lock. Though using a Toptica
ECDL, the laser-head board is replaced to be compatible with the controller from
MOG Laboratories Pty Ltd6. The line width in this configuration can be as low as
400 kHz measured by beating two independent lasers and observing the width of the
beat note on a spectrum analyser. The optimum (minimum) width is obtained when
the grating of the ECDL has been aligned precisely. This also gives a large mode-hop-
free frequency tuning range. All slave laser diode mounts7, temperature controllers8
and laser diode current drivers9 are commercial products. The laser diodes used for
the Zeeman slower laser and absorption imaging slave laser are 30 mW laser diodes
with a centre wavelength of 675 nm available from Toptica. The other diodes are
manufactured by Hitachi10 and have a specified maximum output power of 80 mW
at 658 nm. Some lasers require two optical isolators to eliminate optical feedback.
Figure 4.4 shows the optical setup for imaging and Bragg experiments. The laser
light for Bragg spectroscopy and imaging is offset-locked, which is explained in
appendixA.
4.2 Feshbach coils
In order to investigate strongly interacting Fermi gases, it is necessary to tune the
atoms to the broad Feshbach resonance of 6Li where the scattering length is strongly
enhanced and universal properties can be investigated. Because this Feshbach res-
onance is at 834 G, the magnetic coils, providing the high magnetic offset field, are
5Toptica: DL 1006MOGlabs, model DLC-2027Thorlabs: TCLDM98Thorlabs: TED2009Thorlabs: LDC202 and LDC205
10Hitachi: HL6526FM
74 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
Figure 4.3: Optical setup for cooling. For more details see text.
4.2 FESHBACH COILS 75
Figure 4.4: Optical setup for imaging. To generate the light frequencies for imagingand Bragg experiments, the light is offset-locked to the reference light. ECDL 2:imaging master laser.
76 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
labelled as the Feshbach coils. The set of two coils in Helmholtz configuration is
designed to generate magnetic fields of up to 1.5 kG. They are made of square hollow
copper tubing through which pressurised water is pumped for cooling. Both coils
have 51 turns each and are wound on a mount made from hard plastic. The inner
(outer) radius is 3.5 (5.5) cm with a thickness of 3 cm. The separation of both coils
is 3.5 cm to fit the science glass cell in between. This design requires high currents of
up to 200 A. The magnetic field has its axis along the direction of gravity. The orig-
inal inductance was measured to be 433(10) µH which limits the switching times to
100 ms. The coils are slightly further apart than a perfect Helmholtz configuration,
consequently causing a finite second derivative of the magnetic field. This curvature
was measured and calculated to 0.033(2) G/cm2. In this experiment, 6Li atoms are
trapped in the two lowest high-field seeking states |1/2,−1/2⟩ and |1/2, 1/2⟩, wherethe curvature provides a (anti-) trapping potential in the (vertical y-) horizontal
x, z-plane. With parametric oscillation measurements at 834 G the trapping and
anti-trapping frequencies at 834 G were determined to be, respectively,
ωmag = ωx = ωz = 2π 24.5 s−1 and ωy = 2πi 35 s−1 (4.1)
The magnetic field stabilisation is better than 20 mG over the course of a few days.
The potential bowl of the magnetic trap with frequency, ωmag, is harmonic
Umag =1
2mω2
magz2 (4.2)
and usually dominates the axial confinement of the trap laser beam.
4.3 Dipole traps for evaporation and experiments
In the science glass cell, atoms are evaporatively cooled to create ultracold degener-
ate Fermi gases. Evaporative cooling has been performed elsewhere in high power
optical dipole traps (∼ 140 W) [48], resonant build-up cavities [33], with two differ-
ent hyperfine states [32] or with sympathetic cooling, i.e. using a refrigerant which
can be another isotope [35, 36] or another species [21, 34, 203]. In this experiment,
a single focused gaussian beam from a 100 W fibre laser at 1075 nm is optimised
for evaporation of a single species gas. The power in this primary laser is precisely
controllable through active stabilisation. A secondary 10 W fibre laser at 1064 nm
is used to provide flexible confinements when needed for a particular experiment.
4.3 DIPOLE TRAPS FOR EVAPORATION AND EXPERIMENTS 77
4.3.1 Principle of dipole traps
Since the first realisation of detuned dipole traps for cold atoms [204], they have
become a highly used device for studies on ultracold atoms. A detailed review
on dipole traps can be found in [205]. While cooling in magneto-optical traps is
limited by heating by fluorescent light produced by spontaneous scattering (radiation
pressure, recoil energy), dipole traps are detuned from the atomic transition (far-
off resonant scattering) but still provide a potential in which the atoms can be
trapped. The electric field of the laser light E induces a dipole moment p in the
atom depending on its polarisability α (p = αE). The interaction energy is then
Udip = −1/2 ⟨p · E⟩ and forms the trapping potential which depends on the spatial
profile of the laser intensity I(r)
Udip (r) = −3πc2
2ω30
(Γ
ω0 − ω+
Γ
ω0 + ω
)I (r) , (4.3)
for a given driving laser frequency ω and atomic transition frequency ω0. c is the
speed of light and Γ the natural line width, Γ = 2π 5.9 ·106 s−1 for the D2 transition in
lithium. Red-detuned laser fields have an attractive dipole force which can confine
atoms at an intensity maximum. Spontaneous photon scattering from the dipole
trap is still possible, thus causing heating. The scattering rate is given by
Γdip (r) = −3πc2
2ω30
(ω
ω0
)3(Γ
ω0 − ω+
Γ
ω0 + ω
)2
I (r) (4.4)
and decreases with the square of the detuning while the potential depth scales only
linearly. To minimise heating due to spontaneous scattering, far-detuned dipole
traps are used for trapping atoms.
4.3.2 Single focused dipole trap
The evaporation process is performed in a single focused red-detuned dipole trap.
The atoms accumulate in the vicinity of the focus of the beam. The characterisation
of this trap is approximated by ideal gaussian beam optics. The relevant parameters
are the power P , waist w0 at the 1/e2 minimum radius and the Rayleigh length
zR =πw2
0
λ1075. (4.5)
78 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
Figure 4.5: Schematic of the primary optical dipole laser for evaporation andtrapping (1075 nm) and of the second trap laser (1064 nm). Also shown is theimaging light (red lines).
4.3 DIPOLE TRAPS FOR EVAPORATION AND EXPERIMENTS 79
The intensity distribution for a gaussian beam follows as
I (r, z;P ) = − 2P
πw2 (z)exp
(− 2r2
w2 (z)
), (4.6)
where r2 = x2 + y2 and
w = w0
√1 +
(z
zR
)2
(4.7)
is the beam waist along the beam axis. The trap depth U0 can easily be calculated
from equations 4.3 and 4.6 by setting r = z = 0 and depends on the final laser power
P , so U0 → U0(P ). Trapping of ultracold atoms implies that the thermal energy
of the cold atoms is much smaller than the trap depth. When this condition is
satisfied a harmonic approximation of the optical potential, given by equation 4.3,
can be used which is simply obtained by a Taylor expansion
Udip(r) ≃ −U0 (P )
[1− 2
(r
w0
)2
−(z
zR
)2]. (4.8)
The radial and axial trapping frequencies read at high field
ωr =
√4U0 (P )
mw20
and ωz =
√2U0 (P )
mz2R. (4.9)
The mean trapping frequency is ω = (ωzω2r)
1/3.
The light for the dipole trap laser is provided by a high power multimode fibre
laser from IPG11. The output power reaches 100 W at a centre wavelength of 1075
nm with linear polarisation, a beam quality factor of M2 < 1.05 and a line width
of 3 nm. The beam waist used in our experiments is roughly 42 µm at the centre
of the glass cell, limited by the high laser power used in the experiment. Smaller
waists may result in damage to the glass cell surface as found in a test setup [198].
Evaporation of atoms requires tuning the laser power in a well controlled way,
eventually over three orders of magnitude. The high power range from 100 W down
to 12 W is simply voltage-controlled through an external voltage control of the laser.
Below 12 W, intensity fluctuations may lead to heating of the atoms. Therefore, the
laser power is actively stabilised. A small fraction of the dipole trap light leaking
through a mirror is measured on a photo detector12 after attenuation with a neutral
11IPG: YLR-100-1075-LP12Newfocus: Model 1623
80 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
density filter. The measured voltage is compared to a setpoint signal from the
computer control system and is used to derive an error signal which determines
the amount of radio-frequency power sent by a PID13 controller to an AOM14, see
figure 4.5. The computer control regulates the laser intensities through this AOM
reliably to ∼ 17 mW. More about the trap frequencies of single optical dipole traps
can be found in [189,198].
4.3.3 Crossed dipole trap
While the primary laser is optimised for evaporation, the experiments may need
modified trap frequencies or a different Fermi energy. A versatile tool to manipulate
the trapping potential is formed with a second far-detuned optical dipole trap laser15.
The laser with a centre wave length at 1064 nm has a maximum output power of 10
W and is not actively stabilised. In chapter 6, the secondary laser is overlapped with
the shallow primary trap laser after evaporation. Different ramps down to the final
power of the primary laser, U0 (P ), allow one to change the trapping frequencies of
the combined trap, which in fact varies the Fermi momentum reliably in the range
of interest.
In chapter 7, the ultracold cloud is transferred after evaporation into the second
trap while the primary trap is ramped off. The new trap is deep enough to trap
atoms and heat them to a certain level. In a crossed optical dipole trap, the trap
depth is
Udip =
(2c2P
w2
(Γ1
2ω31
(1
ω1 − ω+
1
ω1 + ω
)+
Γ2
ω32
(1
ω2 − ω+
1
ω2 + ω
)))(4.10)
The harmonic approximation is valid again, and one recovers equation 4.8 for the
axial optical confinement and equations 4.9 for the general trap frequencies.
In order to determine the trap frequencies for the combined trap, the trap
frequencies are measured for 6.8 W at the position of the atoms without the primary
trap. The technique to measure the frequencies is a standard procedure, which is
the observation of the position of an oscillating cloud. In this thesis, trapping and
evaporation is performed at 834 G. The values for the frequencies are ω1064,y =
2π 248 s−1, ω1064,x = 2π 41 s−1 and ω1064,z = 2π 24.7 s−1. The last frequency
includes the magnetic confinement. The waists of the secondary trap are 94.5 µm
in the tight direction and 698 µm in the weak direction.
13SRS: SIM 96014Crystal Tech15IPG: YLR-10-1064-LP-SF
4.4 SETUP FOR BRAGG SPECTROSCOPY 81
Because of the dependence of potential depth of the primary laser on the power
U0 (P ), for a fixed power of the second trap laser, a combination of both traps leads
to the overall mean trapping frequency
ω =(√
ω2r + ω2
1064,y
√ω2r + ω2
1064,x
√ω21064,z
)1/3. (4.11)
The mean trapping frequency yields the Fermi energy ϵF = (6N)1/3~ω and the Fermi
momentum kF =√
2mϵF/~2.
4.4 Setup for Bragg spectroscopy
The principle of Bragg spectroscopy is described in section 3.2.1. The generation of
the Bragg beams is explained in the following and a drawing is given in figure 4.6.
The light is fed from the imaging slave and then passed through an AOM, which
shifts the frequency. After retro-reflection the total frequency shift is 2 × 126 MHz
for the experiments presented in chapter 6 and 2 × 240 MHz in chapter 7. This
AOM is used to trigger the Bragg pulse, which is set to a duration of either 50 µs
or 200 µs. A shutter is necessary to block residual light leaking through the AOM.
The detuning is 2 GHz from the upper excited level, |J = 3/2,mJ = −1/2⟩. This islarge compared to the 80 MHz splitting of the two lowest hyperfine states, |1/2, 1/2⟩and |1/2,−1/2⟩, and the beams couple to each state with a difference of 4 %. Beam
shaping results from coupling into a fibre. The gaussian output is then split into two
paths: one path passes an AOM with a fixed frequency of 80 MHz while the second
path goes through a variable frequency AOM. The difference with respect to the
other beam is set between 0 and 600 kHz, which is called the Bragg frequency. The
setup of the electronic circuit is reported in [189]. A small fraction of each beam
is overlapped on a photo diode. This beat signal is read into a digital oscilloscope
to record the Bragg frequency. The main part of the beams is shone into the cell
where the two beams cross each other at a certain angle θ. The intensity of each
beam is tracked again by photo diodes. The angle determines the imparted Bragg
momentum k = 2kL sin(θ/2), where kL is the laser wave vector at the wave length
λL = 671 nm. The Bragg frequency corresponds to the energy of the probe particle
in a two-photon process. The ratio of imparted momentum to Fermi momentum
is k/kF . This ratio is tunable either by changing k or kF . Changing k requires an
external re-setup of the Bragg beams to vary the angle θ. For convenience kF is
tuned with the power of the primary laser as will be explained in more detail in
section 6.3.
82 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
Figure 4.6: Optical setup for the Bragg frequency generation. Bragg beams 1 and2 have a small frequency difference of 0−600 kHz, which is electronically regulatedwith the LabView voltage via a voltage controlled oscillator.
4.5 DETECTION WITH ABSORPTION IMAGING 83
4.5 Detection with absorption imaging
In order to match the state-of-the-art experimental setups in atom optics research,
advanced imaging is necessary. The improvement of the imaging systems is often
adapted to the specific goals of these experiments, for example imaging in exper-
iments with atom chips, number squeezing, low dimensions, optical lattices etc.,
only to name a few. All the relevant measurements in this thesis are taken with
destructive absorption imaging [206]. Other successful techniques are fluorescent
imaging [207], which requires very high imaging power, or non-destructive phase
contrast imaging [208, 209], where the translation from phase to real intensity can
be rather difficult. A growing number of experimental setups use imaging along
the axial trap direction to obtain images with cylindrically symmetric clouds and
apply a textbook Abel transform to these images, e.g. [210]. Sophisticated setups
accomplish single-atom and single-site resolution with optical molasses induced flu-
orescence imaging via a high-resolution microscope objective [211–214].
For the images here, it is important to obtain true atom numbers and cloud
shapes; for instance in the Bragg experiment, the signal is given by small changes
in the centre of mass of the cloud, and for the temperature calibration the spatial
density distribution must reflect the true shape in the images. The imaging beam
path is in-plane with the dipole trap lasers, and is therefore indicated as side imaging.
In this beam path, there are minimal optical elements between the fibre output and
the CCD camera16. The smaller the number of optical elements, the less likely are
sources of unwanted spatial variations in the beam intensity, which could appear in
images due to interference resulting from multiple reflections from optical surfaces,
or diffraction from aperturing of the beam or from small imperfections in the optics
(dust etc.). The size scale is often near the scale of the imaged object. Previous
imaging was performed from the top where the beam path shared optical access to
the glass cell with the vertical MOT beams. This requires somewhat more optics in
the imaging beam path which degrades the quality of the images noticeably. For this
work, the camera17 for top imaging has been used only to measure trap frequencies
in the weak optical and magnetically confined direction, where one is only interested
in centre of mass motion and not absolute atom numbers.
16Prosilica: GC1380H17Q Imaging: Retiga-Exi-F-M-12-IR
84 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
4.5.1 Imaging hardware
The imaging light is outcoupled from the offset locked imaging slave diode laser and
is shifted by 91 MHz to be resonant with the transition |1/2, 1/2⟩ → |3/2,−3/2⟩.The AOM is amplitude controlled to switch the imaging light on and off. A shutter
is necessary to prevent resonant first-order light leaking through the AOM to the
chamber. The light is sent to the experiment via an optical fibre, see figure 4.5,
which also cleans the spatial mode of the beam. After the cell, magnification is
performed using two 2-inch achromats with focal lengths of 150 mm and 200 mm,
respectively. The 12 bit CCD camera has a chip with 1392 × 1040 pixels, each with
a side length of 6.45 µm. One pixel corresponds to D = 4.85 µm or a resolution of
∆r ≥ 4.85 µm on the camera. Note in this system the diffraction limited resolution
is ∼ 2.5 µm. Absorption imaging requires two pictures, as discussed below. The
shortest possible time interval between the atom and reference picture is desirable
to avoid timescales of mechanical movements or other fluctuations, which can lead
to fringes in the processed pictures. For the images used here, this interval is 60
ms. Some imaging problems can be solved with post-evaluation but ideally the
noise sources should be minimised. This requires a good camera, e.g. with kinetics
mode to minimise the time interval between two pictures, fast readout, etc., and
carefully designed imaging beam paths. If a CCD camera has a limited dynamic
range, clouds with optical depths of the order > 2 can become difficult to image
accurately. Unfortunately, the camera used here is a low budget, non-cooled camera
and has a low quantum efficiency (∼ 0.3). Thus, the imaging noise can only be
reduced to a certain degree.
4.5.2 Image processing
Absorption imaging is based on two images of pixel-by-pixel arrays (x, y), where
one image contains the shadow cast by the atomic cloud B1 (x, y) and the other
serves as a reference picture B2 (x, y). The shadow is highly sensitive to the density
of the cloud. This in turn depends on the photon scattering through the medium.
The standard transmission behaviour including saturation through a medium of
thickness z is described by the Lambert-Beer law [215]
∂I (z)
∂z= −nσ0
α
1
1 + I (z) / (αI0sat)I (z) . (4.12)
I0sat is the saturation intensity and σ0 is the absorption cross section. For the D2 two-
level system in 6Li, the resonant cross section is given by σ0 = 3λ2/2π = 2.14·10−9
4.5 DETECTION WITH ABSORPTION IMAGING 85
cm2, which leads to a saturation intensity of I0sat = 2.55 mW/cm2. The absorption
cross section is reduced by another factor of two, because the imaging laser light
propagates horizontally and thus perpendicular to the quantisation axis of the mag-
netic field provided by the Feshbach coils. In that case, only the σ−-projection of
the linearly polarised imaging light interacts with the atoms. A third dark-image,
Bdark (x, y), obtained when all beams are off, can be taken as a background ref-
erence, to subtract any stray light and CCD dark counts. This is the same for
each experimental run and is subtracted from all the atom and reference pictures,
IA (x, y) = B1 (x, y)−Bdark (x, y) and IR (x, y) = B2 (x, y)−Bdark (x, y), respectively.
In the simplest analysis the shadow image on the CCD chip with respect to the refer-
ence image represents the integrated column atomic density n(x, y) =∫n(x, y, z)dz.
In earlier experiments, the natural logarithm of the ratio of the atom and reference
images led to n(x, y). This assumption is valid for large and dilute samples when
the optical depth, od (x, y) = σ0∫n(x, y, z)dz, is much smaller than unity.
In the following, the optical depth is assumed to take on any values and can
only be found by including the effects of saturation [215]; hence one needs infor-
mation on the incident and final intensities as well as α and Isat, reason being that
imaging a range of clouds from low to high optical densities can hide subtleties,
such as the nonlinear response of the atoms to the probe beam and a reduction
of the effective cross section. However, this change can be corrected by a dimen-
sionless parameter α, which has to be determined from the experiment. By using
strong saturation imaging, the effective cross section is intensity dependent, σ(I),
when I ≫ αI0sat. So, the response of the atoms depends on the effective saturation
intensity Isat = αI0sat.
The optical depth is od (x, y) = σ0∫n(x, y, z)dz with the atomic density n
given by equation 4.12. The optical density represents the information measured in
the absorption image in terms of the density δ0(x, y) = − ln(IA(x, y)/IR(x, y)). In
order to obtain the proper atom number from N =∑
x,y od(x, y)D/σ0, the correct
value of α has to be extracted from
od (x, y) = αδ0 (x, y) +IR (x, y)− IA (x, y)
Isat. (4.13)
where α plays the role of the number correction to the saturation intensity due to the
polarisation of the imaging beam, the structure of the excited state, and the different
Zeeman sublevel populations of the degenerate ground state of the optical transition.
It should be noted that for this thesis large samples of up to 200 000 atoms in one
spin state are imaged and small structures of the clouds are irrelevant. Therefore, a
86 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
large optical depth can be an advantage. Due to the evaluation technique with the
f-sum rule, see chapter 3.8.4, even optical errors such as astigmatism of a tilted lens
can be compensated.
The imaging calibration here is performed according to [215]. To determine
Isat, the imaging power after the fibre is set to a power of Pset = 2.3 mW. The power
of Ipin = 313 µW is measured through a pinhole of 2 mm2 diameter to obtain an
intensity of 9.87 mW/cm2. Dividing this intensity by I0sat = 2.55 mW/cm2 gives a
ratio r = Ipin/I0sat. After taking five reference images at 10 µs at that set power,
Pset, the average pixel count, Rset, is divided by r to yield Isat = 2Rset/r. The
factor of 2 is due to side imaging; as mentioned before the absorption cross section
of the atoms is effectively halved because only the σ−-polarisation of the linearly
polarised imaging beam can drive the atomic transition. Once Isat is fixed, α is
varied in equation 4.13 until the measured atom number is constant over the full
range of intensities used in the calibration procedure. As the imaging beam power
was tuned, the duration of the pulse was varied from 0.96 µs (I ∼ 2.70 mW) to
14.5 µs (I ∼ 0.15 mW) while keeping the number of counts per pixel smaller than a
tenth of the maximum count, i.e. < 400. We find α = 2.2 and Isat = 1510 (counts),
respectively, referenced to a pulse length of ∆t = 10 µs.
This pulse length is a compromise between a maximum possible photon scat-
tering number and minimal smearing effects. Ideally, one would image with an ultra
short pulse and high intensities. This is not practical, and therefore imaging is per-
formed here with ∆t = 10 µs and ∼= 2 mW to minimize recoil blurring and Doppler
shifts: The blurring is introduced by a random walk of the atom after gaining too
much recoil momentum from scattered photons. For our pixel resolution of ∆r =
4.85 µm, a critical number of scattered photons, Np,crit = 3(∆r/∆t)(mλ/(2π~)) ≈12, can be found before recoil blurring starts to play a role [206]. Doppler effects
result from a Doppler shift, where during a given time ∆t an atom has gained a
velocity after absorbing photons from the imaging beam. This new velocity leads to
an effective change in the imaging frequency. When the photon scattering pushes
the atom to a velocity that corresponds to a detuning of ∆ = Γ/2, i.e. is half the
line width, the atom is not really being imaged with resonant light anymore. To
calculate the critical number of scattered photons one can use the expression for the
scattering rate on resonance
Np =Γ
2
I/Isat1 + 2∆/Γ + (I/Isat)
·∆t, (4.14)
For the experimental parameters this leads to Np ≈ 15 at the effective saturation
4.5 DETECTION WITH ABSORPTION IMAGING 87
I/Isat = IR/αI0sat ≈ 2. The imaging power applied for data acquisition is higher
than the saturation intensity but still much less than needed for high-intensity flu-
orescence imaging.
In absorption images, the detection sensitivity is limited by intrinsic photon
shot noise and technical noise. The pictures do not suffer from fringes in general,
but for the discussion of the temperature, see chapter 5, and some measurements of
the temperature dependent contact, see chapter 7, where the result is mainly based
on pixel counts, a fringe removal programme is applied to help reduce photon shot
noise. This shot noise follows a Poisson distribution, where the variance of the pixel
counts is equal to the mean counts per pixel. The noise σN in the real optical depth
in the region of the atoms (ROI) is the square root of the variance of the noise,
σ2R = σ2
A, in picture A and R, which are assumed to have the same noise. Thus,
σN = (D/σ0)√ROI
√2σ2
R, measured in counts. The noise on the dark frame and
the camera readout noise σ2rd are ignored in this analysis. This shot noise can be
reduced with an image algorithm which is discussed in the following section.
Fringe removal algorithm
The basic idea in the fringe removal algorithm is the reconstruction of an optimum
reference image, IQ, from a set of reference and absorption images. The idea is based
on a Gram-Schmidt decomposition and MATLAB18 offers a sophisticated algorithm
for the decomposition [216–218]. In contrast, filtering techniques such as Fourier
filtering use the divided picture and thus can require assumptions about the atomic
distribution. Filtering such as medians also reduces the signal sharpness, which is
important to maintain for Fermi gases where the information is often stored in the
wings.
The following procedure is reported in [219], where here the algorithm is out-
lined. A number k of atom and reference Rk images are taken. MATLAB treats
these as a set of linear vectors and solves them to obtain an orthonormal basis set
of images, where the optimum image IQ =∑
k ckIR,k is the diagonalised version,
i.e. the linear combination of their basis with coefficients ck. The algorithm uses
δ =∑
xmx (IA,x − IQ,x)2, where mx is a mask free of signal and δ is the least square
difference which has to be minimised; explicitly
∂δ
∂cj= 2
∑x
mxIR,x,j
(IA,x −
∑k
ckIR,x,k
)= 0. (4.15)
18Matlab: The MathWorks
88 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
This reduces to ∑k
ckBj,k =∑x
mxIR,x,jIA,x (4.16)
with the matrix elements Bj,k =∑
xmxIR,x,jIR,x,k. This has to be solved for ck with
the solver type ‘LU decomposition’ from MATLAB, where the linear equations are
sorted into lower and upper triangular matrices.
This algorithm is primarily aiming to filter fringes. But in the end, the shot
noise in the optimum image IQ is also reduced compared to a single reference image
IR, the reason being that more than one image is used to construct the reference
picture. The improvement goes with the square root of the number of images and
results in an improvement in the optical density, which for a large number of reference
pictures, σN = (D/σ0)√ROI
√σ2R or equivalently by a factor of
√2 smaller than
the standard absorption image. In other words, the noise of the reference image has
been decomposed in the optimum reference picture and the noise only comes from
the atomic picture.
Other noise sources can also play a role. The atomic cloud may act as a
geometric lens or gradient index lens with an index of refraction. This could lead
to systematics in measurements of the temperature and density. For imaging with
resonant light, lensing effects can be avoided as the refractive part becomes zero.
In most cameras a glass window protects the CCD chip from eventual external
damage. Due to the finite thickness of this glass, the imaging light can undergo mul-
tiple reflections leading to interference fringes in the processed picture (etaloning).
For this reason we remove the glass plate.
Due to the geometry of the imaging beam to trap laser, the cloud is imaged
at a certain angle. For very large clouds, e.g. when the cloud is heated and imaged
after a release time, the cloud has expanded in all directions. But in the horizontal
direction, the cloud size is so big (> zR ≈ 50 µm) that some atoms will lie outside
the depth of focus (±20 µm). When heating atoms, the temperature, a global
parameter, can appear different in the horizontal and vertical directions due to this
focussing. In fact, the temperature seems warmer in the horizontal direction, and
the deviation from the vertical direction becomes worse, proportional to the heating
rate. Therefore, when heating clouds, we rely on the vertical cloud size, so that we
use only the information from the atoms that are in focus.
4.6 SUMMARY 89
4.6 Summary
The production of degenerate quantum gases does not come in commercially avail-
able machines but requires highly customised setups. In order to investigate strongly
interacting Fermi gases, the isotope 6Li is laser-cooled and condensed below the
degeneracy temperature with optical devices in a ultra-high vacuum cell. The op-
tical potentials from a high-power laser set the energy scale of the system, that is
the Fermi energy and momentum for Fermi gases. These quantities characterise
the basic parameters of the experiments explained chapter 6 and 7. The technical
requirements for Bragg spectroscopy are provided by low power and frequency con-
trolled lasers. Opposed to previous theses about this setup, imaging techniques are
considered more carefully. This allows a good quality of pictures free of fringes and
mostly dominated by photon shot noise, camera dark counts and read-out noise.
90 CHAPTER 4. DEGENERATE QUANTUM GAS MACHINE
Chapter 5
Temperature of unitary Fermi
gases
The determination of the temperature in strongly interacting Fermi gases is deli-
cate, from a theoretical as well as an experimental point of view. It is difficult to
find accurate models for the strongly interacting fermionic density profiles at finite
temperature. This chapter is discusses the aspects of thermometry of Fermi gases
and how they are achieved with regard to the results presented in chapter 7.
5.1 Introduction
The study of the thermodynamics of universal properties provides valuable in-
sights into strongly interacting Fermi gases, see e.g. [153, 220–224]. The BEC-BCS
crossover with its naturally strong correlations poses theoretical challenges as most
perturbation theories break down and new are necessary. Determining a universal
temperature scale is problematic since it is difficult to construct a quantitative theory
in terms of a well-defined small parameter and the equation of state is only known
with approximations. The temperature is ideally described by a model-independent
as well as a temperature sensitive quantity. Early theoretical models to explain tem-
perature related data were obtained from the BCS mean field picture [48,225,226].
In general, numerical models need a good comparison with the experiment to find
the appropriate coefficients. The universal relations with the contact parameter
may offer a theory in terms of a well defined small parameter, but these relations
are not easily accessible and do not seem to show a sensitivity to temperature in
the interesting low temperature regime, where the contact changes marginally with
temperature. Appropriate boundary conditions are crucial in the solution of the
problem.
91
92 CHAPTER 5. TEMPERATURE OF UNITARY FERMI GASES
At unitarity, the energy of a strongly interacting Fermi gas deviates from the
ideal gas case over a wide temperature range. Figure 5.4 depicts the energy per
particle of an ideal (green line) and strongly interacting (red solid line) Fermi gas
in the trap, respectively. The energy per particle is calibrated with respect to the
Fermi energy of the non-interacting gas, ϵF . In the unitary case, at temperatures
T > 0.5 TF the curve is nearly a straight line. Below a certain temperature, T ∗,
pairing effects exist but a clear onset of pairing is not visible in the energy of a
trapped gas. Just below T = 0.3 TF , the unitarity energy seems to have a plateau.
At the critical temperature of superfluidity, TC ≈ 0.2 TF , the ideal and unitary gas
saturate due to Pauli blocking. One observes that the unitary energy has here the
biggest deviation from the energy of an ideal gas. The strong interactions lead to
pairing of fermions, and thus bosonic structure, which is most pronounced at low
temperatures.
The experimental determination of the temperature in spin-balanced Fermi
gases at unitarity is of critical importance. The gases are stored in a high vacuum
and cannot equilibrate with an external reservoir of known temperature. Exceptions
are sympathetically cooled Fermi gases, where a bosonic gas acts as an refrigerator
for the spin-balanced Fermi gas, given both are in constant equilibrium. The tem-
perature for bosonic clouds is usually easy to measure. Another exception are Fermi
gases with an imbalanced spin population, where the density of the majority spins,
which do not pair up, behave like an ideal Fermi gas and an accurate fitting model
is applicable.
The most common tool to experimentally access information from fermionic
spin balanced, single species clouds are density profiles obtained from absorption
images. Different models have been developed to determine the temperature of
a Fermi gas though there is no common agreement of a valid analytic model for
these profiles. Lacking an analytic model for the density profiles, fitting of Fermi
statistical distributions lags behind the related fitting of bosonic distributions where
analytic bimodal functions are readily available. When images of clouds are taken,
one observes that attractive interactions lead to a shrinking of the cloud. But at
unitarity the difference in the shape of a balanced interacting and a non-interacting
Fermi gas appears minuscule [112]. The interactions, parameterised by the local
1/(kFa), vary across the cloud and the temperature varies over different cloud shells.
Bosonic clouds below the critical temperature may be accompanied by vortices as a
signature of superfluidity. Signatures of superfluidity in strongly interacting Fermi
gases have been observed [42] but not linked to the temperature, stimulating the
study of the temperatures in strongly interacting Fermi gases.
5.2 IMAGES FOR THE TEMPERATURE CALIBRATION 93
In the following different evaluation procedures for the temperature are com-
pared. First, the processing of the images of heated clouds at unitarity is described
in section 5.2. Then, the determination of the temperature by fitting an empirical
temperature to unitary cloud profiles [48] is explained in section 5.3. The tem-
peratures for the results in chapter 7 are obtained in this way, despite the model
dependent technique. Next, section 5.4 presents a comparison with the NSR theory
by taking the energy per particle from the mean square size of the profiles [50] (and
not from the equation of state as originally proposed). Any deviation is included as
an uncertainty in the temperature acquisition. Another commonly used technique
is described in section 5.5 based on isentropic sweeps of the magnetic field to the
weakly interacting regime, i.e. the unitary gas is mapped onto the BCS regime [50].
In this limit the entropy is conserved and can be made very close to that of an ideal
Fermi gas. Therefore, one data set is taken with adiabatic ramps to 1/(kFa) = −1.5
and the data is compared to the respective NSR prediction. All results are sum-
marised in a final plot, figure 5.4 and compared in section 5.6. Other methods are
also briefly discussed in section 5.7.
5.2 Images for the temperature calibration
In order to heat the clouds, first they are fully evaporatively cooled at 834 G to
create the same conditions and to set the Fermi energy. The mean trap frequency
for this trap is ω = 2π 66 s−1 leading to a Fermi energy of ϵF/~ = ω(6N↑)1/3 =
2π 6.8·103 s−1 for N↑ = 190 000 atoms in one spin state. The optical potential
depth is ∼ 90·103 s−1, which is a little more than 13× ϵF . Experimentally, this trap
allows for the heating and storage of clouds at unitarity up to 1.1 T/TF without any
detectable atom loss.
After evaporation the confining trap is non-adiabatically switched off and
the cloud is allowed to expand for a variable amount of time. The trap is non-
adiabatically ramped up and left on for rethermalisation of the cloud for 400 ms,
which is larger than ω−1 and about 10 times greater than the inverse of the weak
confinement frequency. Five pictures for each release time are taken. In chapter 1 the
density distributions of a Fermi gas are derived and the image process is explained
in section 4.5. Images imply already one integration along the line of sight.
The cloud has an oblate shape and is imaged from the side, i.e. in plane with
the cloud, see figure 5.1. One-dimensional profiles in the horizontal and vertical
direction can be taken from the images. They are normalised before a fitting. In the
vertical direction of the cloud, in-situ images may suffer from optical astigmatism
94 CHAPTER 5. TEMPERATURE OF UNITARY FERMI GASES
x−direction
y−di
rect
ion
10
20
30
40
50
60
70
10 20 30 40 50 60 70
Figure 5.1: Image and integrated profiles in the x- and y-direction for a cold cloudat 834 G after τTOF = 1.3 ms. The units are given in pixels.
(tilted camera lenses) as less than 10 pixels are provided for the profile. Released
clouds are problematic for the x-direction as the camera has a relative angle to
the dipole trap beam of 30.2. Due to the projection on the camera, the cloud is
in reality 2 times larger along the weaker confinement direction than seen on the
camera. After heating, the horizontal direction is easily out of the depth of focus
and the edges of the cloud might be falsely blurred, as discussed in section 4.5.
As a consequence, the temperatures are taken from normalised profiles in the tight
direction (vertical direction, y-direction), which is dominated by the in-focus path of
the cloud. Although, the expansion behaviour, taken from hydrodynamic expansion
equations, can be used to calculate back to the trapped case, adiabadicity in the
switch-off of the trap beam may alter the cloud expansion rate.
For the temperature discussion in this chapter, four data sets are taken: Set I
is imaged at 834 G with a release time of τTOF = 1.3 ms. After this time-of-flight
the optical density dropped to 5.3 (about 580 counts per pixel on the CCD camera).
5.3 EMPIRICAL TEMPERATURE 95
label 1/kFa add TOF techniques
set I 0 - 1.3 ms T , ⟨σ2⟩set II 0 - 0.2 ms T , ⟨σ2⟩set III 0 500 ms hold 1.3 ms T , ⟨σ2⟩set IV -1.5 500 ms ramp 1.3 ms ⟨σ2⟩
Table 5.1: Summary of data taken for the temperature calibration. Five images aretaken for each heating. The evaluation techniques with the empirical temperatureT and mean square size ⟨σ2⟩ are described in the text.
This setting gives reliable counts per pixel but still provides a good signal-to-noise
and enough pixels in the profile for fitting. Images for set II are taken almost in-
situ at 834 G, τTOF = 0.2 ms. For the coldest temperature in this set, the optical
density is about 8.5, including the correction of the saturation intensity, and about
200 counts are left at the peak optical density, which is still much higher than the
background noise of about 16 counts per pixel. In set III the clouds are given an
additional hold time of 500 ms on top of the rethermalisation time and then imaged
after a release time of τTOF = 1.3 ms at 834 G. The 500 ms hold time is necessary
because in a further set IV the cloud is ramped within this time up to 992 G and
imaged at that field after τTOF = 1.3 ms. Then, set III and IV are comparable.
Table 5.1 summarises briefly the different conditions.
5.3 Empirical temperature
One method to determine the temperature at unitarity is based on an empirical
temperature T [48]. The assumption is made that the density distribution of a
unitary gas behaves like an ideal Thomas-Fermi function, which is scaled by an
interaction parameter ξ (0) = 1+β = 0.43 with β ≈ − 0.57 in the zero temperature
limit [37,50,62,83,115,117,118,131,143,152,227–231]. A consequence of universality
on resonance is that the chemical potential of the gas must scale with a universal
function ξ (T/TF ) which only depends on the reduced temperature T/TF [63], such
that µ (r) = ξ (T/TF ) ϵF (r). In the case of T = 0, the chemical potential and density
for a trapped gas are related by
µ (r) = µ0 − V (r) = ξ0ϵF (r) ∝ n2/3 (r) . (5.1)
The exponent 2/3 is exact, and hence the density profile involves the same scaling
with the density as a non-interacting Fermi gas, with a renormalized Fermi tempera-
96 CHAPTER 5. TEMPERATURE OF UNITARY FERMI GASES
ture, see section 1.4.3. However, for finite temperature, ξ (T/TF ) may vary from the
temperature dependence of a non-interacting gas [229]. From fits of these functions
to integrated one-dimensional atomic density distributions, the empirical temper-
ature emerges from one of the fitting parameters [37, 48]. This model-dependent
but relatively temperature-insensitive method has been used to determine the heat
capacity of a strongly interacting 6Li Fermi gas [48] and found to be in accordance
with a pseudogap theory.
Fitting one-dimensional profiles
In the non-interacting case the cloud size has a different characteristic length at
different temperatures. In general, at high temperatures T ≫ TF the gaussian width
σx =√2kBT/mω2
x is correct, but at lower temperatures T ≪ TF the cloud cannot be
smaller than the Thomas-Fermi radius RTF,x =√2ϵF/mω2
x. These quantities affect
not only the temperature range but also the size range. The density distribution
shows a different behaviour inside and outside the Fermi radius
n1D (x) ∝
(1− x2/R2
x)5/2
, x≪ RFx
ex2/σ2
x , x≫ RFx
(5.2)
where σ2x ∝ T but R2
x ∝ ϵF (r). Thermometry for very cold Fermi clouds is somewhat
difficult to derive from the cloud shape as the signature of the temperature relies on
the outer rims of the cloud where the signal-to-noise usually drops.
The three-dimensional momentum distribution at finite temperature follows
from equation 1.37, where the Fermi-Dirac integral is expressed in terms of the
polylogarithmic function, Lin (z) =∑∞
k=1 zk/kn, where z → z(q, x) is complex.
In particular, it follows Lin (z) → −Li3/2 [− exp [(µ− V (r))/(kBT )]] for the three-
dimensional case and
Lin (z) → −Li5/2
[−e
µ−V (x)kBT
](5.3)
for the one-dimensional case, respectively. From the fugacity, exp(µ/(kBT )), one
can define the parameter q = µ/(kBT ), which determines the cloud shape. A small
fugacity (large and negative q) yields the gaussian distribution of thermal clouds. For
a high fugacity (positive q), the clouds approach the zero-temperature distribution
in one direction n1D,0 (1− x2/R2x)
5/2. In the trapped case, where a local density
approximation can be used, the spatial degrees of freedom are integrated out and
the temperature, T , is a global parameter, but the ratio T/TF (r) changes across the
5.3 EMPIRICAL TEMPERATURE 97
shells as the chemical potential, µ(r), changes. Imaging errors, such as out of focus
imaging, saturation etc., can lead to errors in the measurements of T/TF . Signal-to-
noise in the low-density wings of the distribution is a very important factor to obtain
reliable information. In the discussion so far, only one dimensional profiles were
considered which can be used for in-trap pictures or released clouds. The profiles
of released clouds are independent of the trap shape but information about regions
with different local T/TF (r) is integrated out. Two dimensional fits are possible too,
but not used here, for instance, integration over elliptical equipotentials (azimuthal
averaging, Abel transform). This technique has the advantage that the number
of points included in the average grows with the distance from the cloud centre.
Hence, the signal-to-noise is best in the wings. However, these procedures require
large clouds and are very sensitive to the determination of the cloud’s centre position
as well as symmetry.
−150 −100 −50 0 50 100 1500
0.05
0.1
0.15
vertical direction (µm)
norm
alis
ed c
loud
(a.
u.)
all pixelspixels for fitThomas−FermiGauss
Figure 5.2: Extracting cx from fitting to a one-dimensional profile in the verticaldirection of a unitary cold cloud after a release time of τ = 1.3 ms. A Thomas-Fermiprofile at zero temperature is used to fit the circles representing the parabolic innerpart of the cloud. The gaussian fit is for comparison.
The fitting model for the normalised, integrated image profiles is based on an
analytic function for weakly interacting Fermi gases that can be scaled for strongly
interacting Fermi gases as discussed in many textbooks, e.g. [202]. This technique
is applied to sets I, II and III. With the universal constant β = − 0.57, the Fermi
98 CHAPTER 5. TEMPERATURE OF UNITARY FERMI GASES
temperature becomes T ′F =
√1 + βTF , i.e. the temperature is understood in terms
of the empirical reduced temperature√1 + βT = T/TF (r). The density profile reads
then for atoms in one spin state N↑
n (x, T ) = − 6N↑√πRTF,x
(T)5/2
Li5/2
− exp
µϵF
− x2
R2TF,x
T
, (5.4)
which includes the polylogarithmic function Li5/2 and the chemical potential µ. The
first part is left as a fit parameter a and µ/(ϵF T ) is represented as q where q ≥ 0
for T ≤ TC , otherwise q = 0. (The influence of T on q is small since for high
temperatures the chemical potential is a large negative number.) The atom number
can be determined from the profiles. The Thomas-Fermi radius can be calculated
from fixed trapping frequencies:
RTF,x =
√2ϵFmω2
x
=
√2 (6)1/3 ~ωmω2
x
N1/6↑ = cxN
1/6↑ (5.5)
Alternatively, the Thomas-Fermi radius is obtained from the zero temperature profile
n1D (x) =16N↑
5πRTF,x
(1− x2
R2TF,x
)5/2
(5.6)
an we again obtain RTF,x = cxN1/6↑ . In practice, this expression is fitted to the centre
of the coldest experimentally achievable cloud, under the assumption that finite
temperature effects in a Fermi gas mostly affect its wings [232,233]. Both approaches
are consistent in the resulting cx value but as mentioned above, to account for pixel
imperfections, the equation 5.6 is used to provide the Thomas-Fermi radius, see
figure 5.2. With cx fixed, the fit function for the one-dimensional profile becomes
n(x, T
)= aLi5/2
[− exp
(q − x2
T c2xN1/3↑
)]. (5.7)
Temperatures obtained in this way are used later in the presentation of the tem-
perature dependent contact. Figure 5.3 shows two typical profiles which yield tem-
peratures of 0.07 T/TF and 0.52 T/TF . In this representation the uncertainty in
the fitted temperature is given by the standard deviation of five images, which is in
average 0.02 T/TF .
5.4 COMPARISON WITH NSR THEORY 99
−150 −100 −50 0 50 100 1500
0.05
0.1
0.15
vertical direction (µm)
norm
alis
ed c
loud
(a.
u.)
0.07 T/T
F
0.52 T/TF
Figure 5.3: Single shot temperature profiles in the tight direction after 1.3 ms timeof flight at 0.07 T/TF and 0.52 T/TF .
5.4 Comparison with NSR theory
In the low temperature limit T/TF < 1, strong coupling theories are required to
accurately describe the thermodynamics of unitary Fermi gases with strong inter-
actions. In general, any theoretical model to characterise a two-component Fermi
gas at unitarity assumes the s-wave scattering problem and the short range model,
as introduced in chapter 2. A theoretical model must produce a uniform universal
function from which a complete set of universal thermodynamic functions can be
derived. Furthermore, it is more convincing if no fitting functions or adjustable
parameters are required. Strong coupling theories are based on approximate many-
body t-matrix theories, i.e. the ladder structure of Green’s functions with interaction
need to be determined. Noziere and Schmitt-Rink (NSR) described in this way a
normal interacting Fermi gas [162]. In order to include strong pair fluctuations in
the strongly interacting regime, it is necessary to include the ladder sum. The NSR
idea could be fully extended to the below-threshold (< TC) regime, which is referred
to as the GPF (gaussian pair fluctuation) approach [83], to provide a quantitatively
reliable description of the low-temperature thermodynamics of a strongly interacting
Fermi superfluid. The NSR theory uses different assumptions below and above the
critical temperature. Below TC quasi particles are assumed while above TC fermions
are the basic particles. This difference causes discrepancies at TC which can be
interpolated across for trapped clouds. Even though there is a discontinuity in the
100 CHAPTER 5. TEMPERATURE OF UNITARY FERMI GASES
theoretical universal function of the GPF theory, the result for the trapped equation
is much smoother. The NSR theory breaks down at the critical temperature because
so far only two-particle scattering is used. Improvements at the critical temperature
may be achieved with the inclusion of 3, 4 or many-body scattering mechanisms.
Compared to other theories, the NSR predictions show very good agreement with
the experimental data over a wide range of temperatures and different setups [117]
(JILA, ENS, Duke, Rice). The predictions of the NSR theory deviate from those
of mean-field crossover theories [167]. The NSR result for the equation of state is
very accurate at low and high temperatures. At first, the thermodynamic potential,
Ω(µ, T ), is defined with respect to the ideal case, Ω(1)(µ, T ). According to basic ther-
modynamics, their derivatives provide, for instance, the energy E(T ) and entropy
S(T ). These potentials can form a set of universal thermodynamic functions of a
trapped Fermi gas at unitarity. The question is how the thermodynamic potentials
scale with the temperature at unitarity. The large scattering length in the unitarity
limit leaves the interatomic distance as the only relevant length scale. This rescaling
influences the energy and entropy, and leads to the equation of state for an ideal
noninteracting Fermi gas. In the trapped case, one has to make use of the local
density approximation. The atom number is set by N =∫drn(r) with the pressure
dependent particle density n(r) = ∂P (r)/∂µ(r) and ∂µ(r)/∂r = −mω2r where ω is
the mean trap frequency. The rather complicated calculation is performed in terms
of the inverse fugacity [82].
Comparison of the data with NSR theory
The temperatures obtained from the empirical temperature in all sets I-IV can
be compared with the NSR theory if the energy per particle is determined. Five
integrated profiles are averaged before the mean square size ⟨σ2⟩ ≡ ⟨r2⟩ is recorded.In order to reduce noise, ⟨r2⟩ is actually extracted from the polylogarithmic density
distributional fit to the averaged profile. Then, the energy per particle follows
from [50]
E
N↑= 3mω2
i
⟨r2i⟩
(5.8)
where i indicates the respective integration direction. At T = 0, the Fermi energy((6N↑)
1/3~ω)sets the Fermi radius and its re-expression as the mean square size
5.5 WEAKLY INTERACTING LIMIT 101
according to R2TF,i = 8
⟨r2TF,i
⟩; thus
ϵF =1
2mω2
iR2TF,i = 4mω2
i
⟨r2TF,i
⟩(5.9)
Now, the energy per particle with respect to the global Fermi energy for the non-
interacting case is given by
E
N↑ϵF=
3
4
⟨r2i ⟩⟨r2TF,i
⟩ × bi. (5.10)
Since the images are for expanded clouds, the factor bi contains the re-scaling back
to the trapped case (bi =⟨r2i,1.3ms
⟩/⟨r2i,0.2 ms
⟩) as well as to the same units as of⟨
r2TF,i
⟩. Plotting E/N↑ϵF against the temperatures obtained with equation 5.7 allows
a comparison with the NSR theory based on the calculation of the energy of state.
The energy per particle taken in the vertical direction is depicted in figure 5.4 over a
range of temperatures. The data agrees well within the error based on atom number
fluctuations in five shots which is reflected in the calculated in-trap Fermi radius.
5.5 Weakly interacting limit
Another method considers the entropy as a more readily measurable quantity than
the temperature. The entropy, energy, and critical temperature of a strongly in-
teracting degenerate Fermi gas have been derived from the fundamental relation
T = ∂E/∂S of the thermodynamic potentials, i.e. the total energy E and entropy
S [50]. However, the strongly interacting regime has to be mapped adiabatically
onto the weakly interacting BCS limit [51,225], using the knowledge that the mean
square size of the cloud is directly proportional to the entropy for weak interactions.
This method is model-independent due to the fundamental thermodynamic relation
but complicated power-law fitting is necessary to connect the temperature range
above and below the critical temperature continuously [231]. Systematic errors can
occur for non-adiabatic ramping of the magnetic field.
The non-interacting temperature, Ti, of an ideal Fermi gas is accessible from
the density profile. The entropy of a non-interacting Fermi gas is known and is
unchanged in an adiabatic sweep. The isentropic sweep away from resonance tends
to decrease the temperature so that Ti is always somewhat below the temperature
T at unitarity [168, 234, 235]. The remaining question of mapping the temperature
for a trapped Fermi gas at unitarity to Ti is addressed with the NSR theory and
the universal thermodynamic function S(T ). By equating S(T ) and Si(T ), where
102 CHAPTER 5. TEMPERATURE OF UNITARY FERMI GASES
Si is the entropy of an ideal Fermi gas, the temperature T of strongly interacting
Fermi gases is expressed as a function of the non-interacting temperature Ti. S(T )
and Si(T ) must be congruent in the BCS limit. Due to the accuracy with which the
entropy function is ascertained from the equation of state in the NSR theory, the
mapping of the temperatures in the different regimes is well defined [82].
Imaging in the BCS limit
A set of clouds, set IV, is produced by evaporating and heating at unitarity and
then ramping in 500 ms to 992 G (1/(kFa) = − 1.5). Imaging is performed at that
magnetic field after 1.3 ms time of flight. Here, some steps of the programme for the
empirical temperature are used. The Thomas-Fermi density profiles from the fits
are used to obtain the mean square size of the cloud but the fitting parameters are
ignored. Instead, the mean square size is plotted against the temperature obtained
at 834 G (set III), where an additional hold time compensates the ramp time to
reach 992 G. The data agrees well with the NSR prediction at 1/(kFa) = − 1.5.
The ratio of the mean square size at 992 G and 834 G, ⟨σ2992⟩ / ⟨σ2
834⟩, is plotted
against the temperature expressed as E834/NϵF in figure 5.5. The error bars are
the standard deviation of five images. By independently measuring the ratio of the
mean square size, ⟨σ2992⟩ / ⟨σ2
834⟩, and the ratio of the (total) energy at 992 G and
834 G, E992/E834, it can be shown that ⟨σ2992⟩ / ⟨σ2
834⟩ ≈ 1 E992/E834, i.e. the energy
scales with the width not only linearly but with a slope of unity [236].
5.6 Comparison of the methods
The method of fitting Thomas-Fermi profiles to one-dimensional profiles to obtain
the empirical temperature is a relatively quick method and only requires one absorp-
tion image, if cx is given, for instance through a calibration. The standard deviation
of five images is better than 0.02 T/TF . Nevertheless, the model-dependence remains
disputable.
One can also take the mean square size of the cloud and use the calibration
curve at 1/(kFa) = 0 provided by the NSR theory. To find the total uncertainty
from both techniques, we include the standard deviation of five images, δfit, and the
deviation of the empirical temperature from the NSR calibration, δNSR, and add
these in quadrature as δ2tot = δ2fit+δ2NSR. The temperature deviates on average by ±
0.03 T/TF and the total error, δtot, is in agreement with the NSR calibration curve
over the applied temperature range, as shown by the error bars in figure 7.8. The
5.6 COMPARISON OF THE METHODS 103
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
3.5
T/TF
E/N
ε F
1/k
Fa = 0 (NSR)
power law (Duke)1/k
Fa = −1.5 (NSR)
ideal gas (NSR)data at 1/k
Fa = 0
data at 1/kFa = −1.5
Figure 5.4: Presentation of the temperature calibration. Black dots are datapoints from unitarity fits and are to be compared with the NSR curve for 1/(kFa)= 0 (red line). Adiabatic sweeps to 1/(kFa) = − 1.5 yield the blue dots and are tobe compared with the NSR curve for 1/(kFa) = −1.5 (blue line). Also plotted arethe ideal gas case (green line) and the power law obtained by the Duke group [231](red dashed line). Note that all error bars on the data points are only the standarddeviations from the empirical temperature.
temperature measurement performed by the ENS group [130, 173] agrees with the
NSR curve within 1 % [167].
For comparison the clouds are imaged at 1/(kFa) = − 1.5 after they are heated
at unitarity, to see if the temperature obtained at unitarity can be reproduced
after an adiabatic sweep to the weakly interacting regime, see blue data points in
figure 5.4. The temperature obtained from set III is compared to the temperature
read off the NSR calibration curve at 1/(kFa) = − 1.5 using again δ2tot = δ2fit+δ2NSR.
In the tight direction of the cloud the relative error is better than 5 % or 0.03 T/TF
on average. The lowest two energy data points are excluded since they may have
been underestimated due to a high optical density.
104 CHAPTER 5. TEMPERATURE OF UNITARY FERMI GASES
0.5 1 1.5 2 2.5
1
1.1
1.2
1.3
1.4
1.5
E834
/NεF
σ2 992/σ
2 834
Figure 5.5: Ratio in the y-direction of the mean square size in the weakly interact-ing regime, σ2
992 ≡ ⟨σ2992⟩, to the mean square size at unitarity, σ2
834 ≡ ⟨σ2834⟩, plotted
against the temperature expressed in terms of the ratio of unitarity energy, E834/N ,per particle to the Fermi energy, ϵF . The data is in qualitative agreement with [50].
5.7 Alternative approaches
It has also been suggested to adiabatically convert the unitary gas to the deep
BEC limit (isentropic thermometry), where a subsequent measurement of the BEC
temperature T ′ of a resulting Bose gas can be taken [234]. Such techniques have been
used [41], but the conversion between T ′ and the unitary temperature T remains
model dependent. Moreover, a side effect of sweeping to the deep BEC side is heating
due to the adiabatic formation of molecules and a subsequent molecular decay.
An alternative may be to use the information about the temperature from the
equation of state, P (µ, T ). The experimentally derived energy per particle [173]
and theoretically predicted values [235] are found to be in good agreement over the
range 0 − 0.8 T/TF . The theoretical model uses the NSR scheme to calculate the
equation of state to obtain measurable quantities such as the entropy and chemical
potential. For the experimental data, sympathetic cooling of 6Li by 7Li was used.
In-situ images of doubly integrated density profiles lead, as proposed [237], to the
equation of state of a homogeneous Fermi gas of 6Li atoms. This requires the
measurement of the local pressure P (µ(z), T ) or the local thermodynamic potential
Ω(µ(z), T ) = −P (µ(z), T )δV in a small volume δV at position z, where the chemical
potential follows from the local density approximation µ(z) = µ0 − V (z) and the
5.8 SUMMARY 105
central chemical potential µ0. Independently, the temperature in those experiments
by the ENS group was determined from bosonic 7Li which played the role of a
refrigerant in the cooling cycle [173].
At high temperatures T/TF ≥ 1, gaussian fitting is accurate to within 1 %. The
fugacity, z = exp (µ/kBT ), is much smaller than unity as the chemical potential µ
converges to −∞. Thus, the fugacity is a small parameter and the virial expansion is
an appropriate model, where the thermodynamic potential of the interacting Fermi
gas can be calculated with a cluster expansion in powers of the fugacity [116, 169].
The temperature independent virial coefficients ∆bn = bn − b(1)n appear as the nth
prefactors of the polynomial function of the fugacity:
Ω− Ω(1)
V= −2kBT
λ3[∆b2z
2 + ...+∆bnzn + ...
]. (5.11)
The thermal deBroglie wave length is given by λ ≡ [2π~2/(mkBT )]1/2
and the volume
as V . Note that the difference Ω− Ω(1) is quantum cluster expanded. At unitarity,
the virial coefficients are known up to fourth order. The virial expansion can bridge
many-body and few-body systems in the high temperature limit. In a trapped gas
the virial expansion can work well down to T ≈ 0.5 TF [169].
When the fugacity and the virial expansion coefficients are known, then it
should be possible to find the energy and the density profile for the trapped case.
One has to make use of the local density approximation to calculate the quantities
in the trapped cloud. A series of fugacities would then enter the polylogarithmic
function. With knowledge of the virial expansion coefficients, this would provide a
straight forward method to obtain analytic density profiles for temperatures down
to T ≤ 0.5 TF .
5.8 Summary
Strongly interacting Fermi gases with a balanced spin mixture are prepared in an
ultra-high vacuum chamber, and conventional thermometry via an externally equi-
librated refrigerant is not possible. Absorption images yield the spatial density
distribution, from which the temperature is extracted. Accurate expressions for the
density profiles are difficult to define for unitary Fermi gases. So far, the solution
is a shape comparison with the noninteracting Fermi gas together with a rescaling,
since the interactions shrink the cloud. In order to determine the temperature, these
modified Thomas-Fermi profiles are applied. For comparison, the mean square size
as a model-independent quantity is related to the energy per particle. The NSR
106 CHAPTER 5. TEMPERATURE OF UNITARY FERMI GASES
theory provides a calibration curve that relates the energy to the temperature and
therefore the temperatures can be read off if the mean square size is given. There-
fore, the two methods determine the temperature from the density distribution of
a strongly interacting Fermi gas. The empirical temperatures agree with the NSR
calibration within 5 %.
Chapter 6
Universal behaviour of pair
correlations
Universality in ultracold strongly interacting Fermi gases is studied by stimulated,
two-photon inelastic Bragg scattering for a range of energy and momentum transfer.
The high momentum transfer allows a spectroscopic measurement of pair correla-
tions and the high momentum distribution of the gas. At unitarity, the short-range
physics is governed by contact interactions. The relevant universal parameter is the
contact, I, which connects microscopic and macroscopic properties as discussed in
chapter 2. The contact depends on the dimensionless interaction strength 1/kFa
and the relative temperature T/TF ; the latter is discussed in chapter 7. This chap-
ter provides the basic understanding for the static structure factor from which the
contact can be determined. The static structure factor and its universal behaviour
is measured in the unitarity regime.
6.1 Introduction
Strongly interacting Fermi gases provide a paradigm to study properties such as su-
perfluidity or universality [35,44,48–51,117], as described in chapter 1. The quantity
considered here to investigate universality is the static structure factor. The static
structure factor quantifies macroscopically short-range pair correlations, which are
difficult to measure directly. With the help of Tan’s universal relations it can be
shown that the static structure factor follows a simple universal law. Universality,
the contact, I, and the Tan relations have been introduced in chapter 2. Recall that
the physical background, on which the universal relations are built, assumes that the
zero range scattering length a exceeds the interparticle separation r which greatly
exceeds the short range of the interaction potential r0. Starting with the universality
107
108 CHAPTER 6. UNIVERSAL BEHAVIOUR OF PAIR CORRELATIONS
of the static structure factor in section 6.2 its applicability to Bragg spectroscopy is
described. Experimental verification in section 6.3 contains the details on how the
Bragg spectra are obtained. With the basic theory of linear response theory from
chapter 3, the sum rules can now be applied. In this context, section 6.4 presents
how the interaction dependence of the contact across the crossover is extracted from
previous work of this group [80].
6.2 Universality of the static structure factor
In a quantum fluid, short-range structure depends upon the relative wave function
of the interacting particles, ψ↑↓ ∝ 1/r − 1/a. Tan showed that the pair correlation
function, g(2)↑↓ , diverges as I/r2 at short relative distance between two particles,
r < 1/kF , and may be written as
g(2)↑↓ (r) =
I16π2
(1
r2− 1
ar
), (6.1)
This universal law is related to the (microscopic) correlation function, as discussed in
equation 2.9. According to Tan, the contact I quantifies the likelihood of finding two
fermions with opposite spin close enough to interact with each other. I embodies all
of the information necessary to determine the many-body properties, i.e. it depends
on the density n ∝ k3F , the interaction strength 1/kFa and temperature T/TF of
the system. The two-body correlation function g(2)↑↓ is usually not easy to access
experimentally. Hence it is necessary to measure macroscopic quantities with a well
defined dependence on the correlation function.
The static structure factor, S(k), for example, is a macroscopic quantity and
depends on the two-body correlation function through its Fourier transform, see
equation 3.14,
S (k ≫ kF ) = 1 + n
∫dr[g(2) (r)− 1
]ei~k·r. (6.2)
In general, two particles can have parallel or antiparallel spin, so the total static
structure factor is
S =S↑↑ + S↓↓
2+S↑↓ + S↓↑
2. (6.3)
For a spin balanced gas, the number of spin-up and spin-down atoms is equal, and
thus S↑↑ = S↓↓ and S↓↑ = S↑↓ lead to S = S↑↑ + S↑↓. Furthermore, for large
6.3 EXPERIMENTAL VERIFICATION 109
momentum transfer, q = ~k ≫ ~kF , e.g. a spin-up particle only scatters into states
with a preoccupied spin-down particle or an empty state due to the Pauli exclusion
principle. Hence, the autocorrelation of parallel spins goes to unity, g(2)↑↑ (r) → 1, and
consequently in equation 6.2 the static structure factor approaches unity, S↑↑ → 1.
This means that for a spin balanced Fermi gas only the correlation between particles
with composite spins contribute to the two-body correlation function [75,82],
S(k ≫ kF ) = 1 + S↑↓. (6.4)
The Fourier transform of equation 6.1 is explicitly [83]
S↑↓ (k ≫ kF ) =I
4NkF
kFk
[1− 4
πkFa
(kFk
)]. (6.5)
The total static structure S(k ≫ kF ) = 1 + S↑↓ can be measured with Bragg
spectroscopy, from which S↑↓ is extracted. In Bragg spectroscopy one probes the
linear response of instantaneous fluctuations in the gas. Signatures of correlated
pairs and free particles are represented in the dynamic structure factor, S(k, ω) [75].
In fact, the response can be quantified by measuring either the momentum [78] or
energy transferred by a Bragg pulse. Integration over the S(k, ω) gives S(k) which
represents in turn the two-body correlation function through the Fourier transform,
see equation 6.5. The static structure factor S(k) derived in chapter 3 follows a
universal law (to first order). Equation 6.5 shows that the ratio I/NkF follows a
universal scaling, that is, the universal character is understood as a dependence
solely on the dimensionless interaction strength via the inverse scattering length a,
the relative probe momentum according to the short-range interparticle distance
1/r << 1/r0 ∝ k/kF , and the dimensionless universal contact parameter I/NkF .In theory this expression is universally applicable to any dilute strongly interacting
Fermi system with contact interactions after a rescaling. The expression is also
model independent which opens the possibility to compare the observed mechanisms
with other areas in physics, e.g. crystallography or solid state physics.
6.3 Experimental verification
Bragg spectra reflect the composition of the gas, being dominated by bosonic mole-
cules below the Feshbach resonance, pairs and free fermionic atoms near unitarity,
and free fermions above resonance. The Bragg spectra are reconstructed in the inset
of figure 6.7a. After the Bragg pulse, the distorted cloud shows the effects of the
110 CHAPTER 6. UNIVERSAL BEHAVIOUR OF PAIR CORRELATIONS
underlying correlations. The revealing of these correlations can be complicated by
elastic collisions between scattered and unscattered particles [238], which are hard
to separate in the BEC-BCS crossover. Nonetheless, elastic collisions do not present
a major problem as they preserve centre of mass motion.
In this section it is shown how the universal behaviour of the static structure
factor is verified over a wide range of probe momenta k/kF as well as at different
interaction strengths 1/kFa. The measurements are compared to new calculations
for the contact based on a below-threshold Gaussian pair fluctuation theory [82].
The general trend of S(k), and therefore the two-body correlation function, is a
smooth increase with both lower probe momenta and higher 1/kFa on passing from
the BCS to the BEC side. Equation 6.5 forms the heart of this chapter and describes
the short-range pairing on both sides of the Feshbach resonance. An equivalent
interpretation identifies this result as a measurement of the interaction dependence
of the contact for a spin balanced gas in the zero temperature limit, as previous
theoretical [70,83,84] and experimental [36, 133] works have studied.
The experimental proof of the universal character of equation 6.5 up to first
order requires an independent variation of the probe momenta k/kF over a wide
range at different values of the interaction parameter 1/(kFa). For different combi-
nations of these parameters, Bragg spectra of ultracold Fermi gases are taken with
the momentum transfer method as described in chapter 3 to relate the centre of mass
displacement, ∆X, to S(k, ω) and S(k). These investigations take place in strongly
interacting Fermi gases, so there are inevitable collisions between scattered and un-
scattered particles after the Bragg pulse which alter the atomic distribution. But
these collisions preserve the centre of mass momentum and therefore measuring ∆X
still provides an accurate measure of the momentum transferred regardless of the
spatial details of the cloud. This technique was found to be adequate over the whole
BEC-BCS crossover regime where the collisional details change [80]. The shape of
the spectra is non-trivial. For that work, k/kF = 5 and T/TF ≈ 0.1, and spectra
have been calculated with a method based on a random phase approximation [192].
In this chapter, the data evaluation relies completely on the experimental data and
no fitting is applied.
6.3.1 Procedure
The experimental sequence starts with the preparation of an ultracold sample of6Li atoms which is loaded from a magneto-optical trap into a single optical dipole
trap (λ = 1075 nm). Evaporative cooling at a magnetic field of 834 G leads to a
6.3 EXPERIMENTAL VERIFICATION 111
Figure 6.1: Optical Bragg beam setup for the verification of the universal staticstructure factor.
resonance superfluid containing N ≈ 3× 105 atoms in an equal mixture of the two
lowest lying ground states |F = 1/2,mF = ±1/2⟩ at T ≤ 0.1 TF .
To gain more control over the trapping frequencies of the cloud, a second far
detuned laser (λ = 1064 nm) is ramped up adiabatically to intersect with the first
trapping beam at an angle of 74. The configuration of the trap and Bragg beams
is depicted in fig.6.1. In this crossed beam configuration both laser intensities are
controlled individually such that the mean harmonic confinement frequency of the
crossed trap ω can be tuned over the range ω = 2π × (38 → 252) s−1. At the
same time the aspect ratio ωx,y/ωz varies from 3.4 to 16, so the cloud is always in
the three-dimensional regime. Consequently, the control over the trap frequencies
determines the Fermi wave vector kF according to kF = (48N)1/6√mω/~, which is
explicitly kF = 2.1 µm−1 → 5.3 µm−1. As the length scale is related to the atomic
density n(r)2 = 6πk3F , the change in trapping frequencies translates to a change in
the relative momentum k/kF = 3.5 → 9.1. Figure 6.2 shows graphically the range
over which the ratio k/kF can be varied with the laser powers of the primary trap.
Next, the scattering length a is addressed with an adiabatic ramp to the final
magnetic field allowing the dimensionless interaction strength, 1/(kFa), to be tuned
in such a way as to keep the product kFa constant. Note that the parameters kF
112 CHAPTER 6. UNIVERSAL BEHAVIOUR OF PAIR CORRELATIONS
Figure 6.2: Ratio of theBragg momentum, to theFermi momentum, k/kF ,as a function of the powerof the primary dipole traplaser.
0 50 100 150 200 250 3002
3
4
5
6
7
Primary trap laser power (mW)
k/k F
and a are independent ’control knobs’ whereas 1/(kFa) is only constant for a chosen
kF if the respective scattering length a is fine-tuned with the magnetic field, B.
Assuming 140 000 particles in one spin state, for 1/kFa = + 0.3, the magnetic field
is changed from B = 802.8 G at 2π 250 s−1 and kF/k = 0.284 to B = 820.0 G at
2π 44 s−1 and kF/k = 0.119. For 1/kFa = − 0.2, the magnetic field is changed
from B = 860.2 G at 2π 252 s−1 and kF/k = 0.285 to B = 844.5 G at 2π 45 s−1
and kF/k = 0.120. After equilibration at the final magnetic field for τ ≫ 10 ω−1,
the Bragg pulse (PBr = 40 µW) is applied onto the trapped cloud for 50 µs, which
is short compared to the two-photon Rabi cycling period, ensuring that spectra
are obtained in the linear response regime. The imparted momentum is k/kF with
fixed k = kL sin(θ/2), that is, with a constant angle between the two Bragg beams,
and the Fermi momentum kF is tunable. The recoil energy ωR = 2~k2/m for one
atom with mass m is defined by the angle of the Bragg beams. For this setup ωR
= 2π 300·103 s−1. To enhance centre of mass signal the cloud travels for τtrap =
1 ms in the trap and expands further for τTOF = 2 ms after turning off the trap.
The imaging frequency is tuned to be resonant with the final magnetic field. The
trapping frequency along the direction of the Bragg beam is 2π 24.5 s−1, which is the
weakest confinement. Therefore, one has a quarter of the corresponding oscillation
time, ≤ 10 ms, before the cloud turns around. This procedure is carried out for
every Bragg frequency ω = 0− 2π 430·103 s−1. The process from single pictures to
S(k, ω) is described in the following.
6.3 EXPERIMENTAL VERIFICATION 113
6.3.2 Creating Bragg spectra
From the images, line profiles and their first moment are extracted using the fact
that the transferred momentum is directly proportional to the resultant ∆X(k, ω),
see chapter 3. A graphic guide to the procedure is depicted in figure 6.3. The vertical
axis is labelled as x, and represents the direction of the applied Bragg pulse. The
horizontal, non-Bragg direction in the image is labelled as y. Each image i yields
a two-dimensional atomic distribution, ni(x, y), and is taken for a different Bragg
frequency ω.
For every image i where the Bragg frequency is varied, a one-dimensional sum
is taken over the y-direction to obtain a line profile in the x-direction, ni(x). The
region of interest in the x-direction is chosen to be as small as possible to include all
atoms but exclude unnecessary background that only adds noise. After summation
the area under each line profile ni(x) is normalised to unity. From the line profiles
ni(x) the first moment, labelled ’(1)’, is evaluated from
X(1)i =
∑x ni(x) · x∑x ni(x)
. (6.6)
Hence, X(1)i denotes the centre of mass taken from the line profile of each image i.
X(1)(ω) ≡ X(1)i represents a spectrum where the centre of mass changes with the
applied Bragg frequency, ω.
Since the interest lies in the centre of mass displacement, it is necessary to
determine a reference position of an unperturbed cloud, denoted as X(1)off . This is
the equivalent of taking the centre of mass position following an off-resonant Bragg
pulse. The offset X(1)off is subtracted from all the X(1)(ω). Now the quantity of
interest is
∆X(1) = X(1)(ω)−X(1)off . (6.7)
Derivation of the dynamic structure factor
To compare Bragg spectra, the dynamic structure factor is normalised to the atom
number N with the definition for the f-sum rule, described in equation 3.54. The
f-sum rule states that the integration over the energy times the dynamic structure
factor is proportional to the atom number. Hence, the normalised dynamic structure
factor is obtained from the centre of mass displacement using
P∆X(ω) =∆X(1)(ω)∑
ω ∆X(1)(ω) · ω
. (6.8)
114 CHAPTER 6. UNIVERSAL BEHAVIOUR OF PAIR CORRELATIONS
-y
(a)
6
x
(b)
Bragg frequency (ω/ωR)
x (p
ixel
)
0.5 1 1.5−50
0
50
100
(c)
(d)
6
x
(e)
0.5 1 1.50
5
10
15
Bragg frequency (ω/ωR)
Cen
tre
of m
ass
disp
lace
men
t (pi
xel)
(f)
Figure 6.3: From images to line profiles to spectrum: (a,d) unperturbed cloud.(b,e) Scattered cloud at the recoil frequency. (c) Integrated line profiles for all im-ages. (f) Centre of mass displacement (points) spectrum obtained from line profiles.The line is a guide to the eye.
6.3 EXPERIMENTAL VERIFICATION 115
Finally, the real dynamic structure factor is for any momentum k
S∆X (k, ω) =P∆X(ω)
1− e−~ω/kBT, (6.9)
which takes into account finite temperature through detailed balancing. For the
measurements that follow in this chapter, the temperature at unitarity is ∼ 0.1 T/TF
and the denominator approaches unity in the relevant frequency range.
6.3.3 Dynamic response
Being now able to control the relative probe momentum k/kF and the interaction
strength 1/(kFa), a sequence of energy resolved Bragg spectra can be taken for any
parameter set. Figure 6.4 shows Bragg spectra for various probe momenta sorted by
1/(kFa) = 0.0, + 0.3, − 0.2. At high momenta, the pair and free-atom excitations
are clearly distinguished at frequencies of ωR/2 ∼ 2π 150·103 s−1 and ωR ∼ 2π
300·103 s−1, respectively. With decreasing probe momenta, the signal strength at
the pair frequency (ωR/2) increases. The signal at pair frequencies is generally
increasing from 1/(kFa) = − 0.2 to + 0.3 until the bound state is reached on the
BEC side (bound molecule regime).
These spectra represent the dynamic structure factor of the strongly interacting
Fermi gas. Probed with Bragg spectroscopy, they respond to single atom and pair
excitations. The height and sharpness of the pair peak is commensurate with the
condensate fraction. At unitarity, it indicates a substantial fraction of correlated
pairs. At 1/(kFa) = − 0.2 (BCS side), the pair size is larger than on the BEC
side [56]. The spectra, and therefore S(k, ω), are continuous and more dominated
by incoherent response to the density fluctuation ρ†~k, see chapter 3. The pair peak
at ωR/2 is considered as a direct scattering of pairs and not as a collective mode
that responds coherently and resonantly to that same excitation. As a result, one
does not expect a discrete spectrum.
In this experiment, single pair excitations in S(1) (k, ω) are most likely as they
are obtained by exciting from a momentum state p ≤ ~kF to a state (p+ ~k).Because of the exclusion principle, |p+ ~k| must be larger than ~kF . The availablevalues of p are exactly the same as for a noninteracting Fermi gas (chapter 3.3), i.e.
the values of p are situated between the Fermi sphere SF and the sphere SF shifted
by −q. The excitation energy of the particle-hole pair is ϵp+q − ϵp. In the case of
q > 2~kF and as seen in these spectra, the single pair spectrum does not start from
ω = 0, but from some finite value of ω. For q < 2~kF , the spectrum of collective
excitations would have covered a finite range of energy, starting linearly with ω from
116 CHAPTER 6. UNIVERSAL BEHAVIOUR OF PAIR CORRELATIONS
Figure 6.4: Spectra fromthe centre of mass displace-ment for different kF/k anddifferent interaction strengths,1/(kFa) = 0.0, − 0.2 and + 0.3.Clear signals of atomic responseare seen at the recoil frequencyωR and of pair response at ωR/2.The higher kF/k the more pairsignal emerges. From the BCSto BEC side, a) to c), the pairsignal becomes stronger until onthe BEC side bound moleculesdominate over the atomic re-sponse.
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
ω/ωR
S(k,
ω)
(ωR
/N)
k
F/k
0.120.170.230.29
(a) 1/(kFa) = − 0.2
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
ω/ωR
S(k,
ω)
(ωR
/N)
k
F/k
0.120.170.220.29
(b) 1/(kFa) = 0.0
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
ω/ωR
S(k,
ω)
(ωR
/N)
k
F/k
0.120.170.220.27
(c) 1/(kFa) = + 0.3
6.3 EXPERIMENTAL VERIFICATION 117
zero to some maximum value plus a sharp cut-off [76].
The spectra shown in figure 6.4 are selected spectra for better visibility. The
whole set of spectra for all interaction strengths, 1/(kFa) = + 0.3 (BEC), 0.0 (uni-
tarity), − 0.2 (BCS), can be plotted in a two-dimensional false colour representation.
The vertical axis represents the energy in terms of the frequency ω/ωR and the hor-
izontal axis denotes the inverse of the relative probe momentum, kF/k. The more
signal of either pairs or atoms the higher is the intensity (bright yellow). Black
areas correspond to off-resonant response. In the very high momentum limit, i.e.
kF/k → 0, response of single atoms is expected at ω/ωR = 1, while at lower mo-
menta, kF/k → + 0.5, which is equivalent to k/kF = 2, the pair response becomes
more significant [75]. On the BEC side, response at the pair frequency is visible
for all momenta. At unitarity, the single-particle response is stronger at a high
momentum transfer, but there is significant weight at the pairing frequency at low
momenta. On the BCS side, the atomic signal is most pronounced but still shows
some tendency to ω/ωR = 0.5. The blue dots depict the frequency referring to the
centre of mass of the spectrum. In first approximation a linear slope can be fitted
through the blue dots. The value of the slope depends on the interaction strength,
1/(kFa). The slope is a fit of the type 1 + s · kF/k, giving s = − 1.82 (BEC), −1.43 (unitarity) and − 1.13 (BCS), respectively. This plot is reminiscent of the cal-
culation of the effective mass of the band structure in solid state physics, where the
effective mass, m∗, follows from m∗ = ~2 (d2E/dk2)−1and the energy E. Despite
the resemblance, clear parallels cannot be made. For a start, a parabolic fit shows
difficulties. Nevertheless, the frequency according to which the centre of mass of
the whole atomic distribution occurs can be related to an energy of the effective
mass, which changes with the excitation momentum. This leads to the well-known
assumption that for infinitely low excitation momenta, k/kF ≪ 1, collective modes
can be excited. The effective mass, m∗, is related to the heat capacity and the spin
susceptibility.
6.3.4 Static response
At the beginning of this chapter, equation 6.5 was introduced stating that with
knowledge of the contact I/NkF it is theoretically possible to measure an exact
quantity in a unitary Fermi gas. One only needs to measure the static structure
factor accurately. A powerful tool to unambiguously determine S(k) on the exper-
imental side is given by sum rules as explained in section 3.8.4. S(k) follows from
the zero moment weighted S(k, ω), while the normalisation to the atom number is
118 CHAPTER 6. UNIVERSAL BEHAVIOUR OF PAIR CORRELATIONS
0.15 0.2 0.25
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
ω/ω
R
kF/k
BEC
0.15 0.2 0.25
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
kF/k
unitarity
0.15 0.2 0.25
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
kF/k
BCS
Figure 6.5: Illustration of S(k, ω) as a false colour representation of the signalstrength for different probe momenta and different interaction strengths, 1/(kFa) =+ 0.3 (BEC), 0.0 (unitarity), − 0.2 (BCS). The frequency according to the centreof mass of the spectrum is given by the blue points. The slope is a fit of the type1 + s · kF/k, giving s = − 1.82, − 1.44 and − 1.13, respectively. For infinitelyhigh probe momenta ω/ωR = 1, i.e. excitation occurs only into free atomic states.S↑↓(k) measures the pairing strength and approaches 0.5 ω/ωR for lower momenta.An open question is if this is the limit of measuring the effective mass of collectiveexcitations in the low momentum regime with kF/k ≥ 1. White dashed lines are aguide to the eye.
6.4 INTERACTION DEPENDENCE OF THE CONTACT 119
obtained from the first moment weighted S(k, ω) (f-sum rule). This procedure is
model-independent and does not require any details of the experimental parameters.
The total static structure factor therefore reads according to equation 3.54
S∆X (k) =
∑ω S∆X (k, ω)∑
ω S∆X (k, ω) · ω· ωR ∝
∑ω ∆X (k, ω)∑
ω ∆X (k, ω) · ω· ωR (6.10)
where ~ωR is the recoil energy. One simply integrates over the zeroth and first
moment of the energy resolved Bragg spectra to obtain the total static structure
factor. In order to extract S↑↓(k), one can use equation 6.4.
For every interaction strength 1/(kFa) = − 0.2, 0.0, + 0.3 the static structure
factor S↑↓(k) can be represented as a function of k/kF according to equation 6.5.
Figure 6.6 summarises without any free parameters how the experimental data ob-
tained from the Bragg spectra compare to the theoretical prediction of equation 6.5,
where I/(NkF ) is obtained from calculations [82] in the zero temperature limit.
From equation 6.5 it is clear that at unitarity (a→ ∞) S↑↓(k) is expected to
vary linearly with kF/k. The first order corrections with decreasing probe momenta
cause a downwards correction for positive 1/(kFa) and an upwards correction for
negative 1/(kFa). Within the experimental uncertainties the experimental data
agree well with the theory displaying a downward curvature for 1/(kFa) = + 0.3
and upward curvature for 1/(kFa) = − 0.2 due to the first order term. The fitted
slope of 0.77±0.07 for 1/(kFa) = 0 is slightly below the zero temperature prediction
of 0.81 which is most likely due to reduced pairing at the finite temperature (T/TF
= 0.10 ± 0.02 at unitarity). The next chapter confirms this reduced pairing. This
temperature effect is less pronounced for the other two sets. At 1/(kFa) = − 0.2 the
temperature will be lower following the adiabatic magnetic field sweep [234] while
on the BEC side at 1/(kFa) = + 0.3 the temperature will be higher but pairing
persists at much higher temperatures.
6.4 Interaction dependence of the contact
In this section, the interaction dependence of the contact across the Bose-Einstein
condensate (BEC) to Bardeen-Cooper-Schrieffer (BCS) crossover is discussed for a
single momentum transfer and low temperatures. The contact characterises in an
elegant way the smooth decrease of pair correlations when a Fermi quantum fluid
crosses over from BEC to BCS superfluids. There have been a number of experi-
ments that measured the interaction dependence of the contact. The contact C was
first extracted [70] from the number of closed-channel molecules determined through
120 CHAPTER 6. UNIVERSAL BEHAVIOUR OF PAIR CORRELATIONS
0 0.05 0.1 0.15 0.2 0.25 0.3 0.351
1.1
1.2
1.3
1.4
kF/k
S(k
)
1/(k
Fa)
− 0.2 0.0 + 0.3
Figure 6.6: Verification of equation 6.5 as measured and calculated static structurefactor versus kF/k for 1/(kFa)+ 0.3, 0.0, − 0.2. The relative probe momentum k/kFis varied by changing the mean trapping frequency ω. Vertical error bars are dueto atom number fluctuations and uncertainties in measuring the centre of mass andhorizontal error bars are due to atom number fluctuations and uncertainties in ω.Solid lines are the zero temperature theory and the dashed line is a straight line fitto the 1/(kFa)+ 0 data yielding a slope of 0.77 ± 0.07. For better visibility of thedownwards and upwards curvature the straight dash-dotted lines indicate the casewithout the first order correction of the theoretical prediction.
photo-association [36]. Also, in agreement with [70], the contact was measured by
comparing the momentum distribution of a ballistically expanded cloud, the high-
frequency tail of a radio-frequency spectrum, and the momentum distribution from
photo-emission spectroscopy [133]. The same work verified experimentally the adi-
abatic and virial Tan relations which are interpreted with regard to thermodynamic
relations. The contact density was determined from the equation of state for a
homogeneous gas [130].
The interaction dependence of S(k) across the Feshbach resonance has been
measured with Bragg spectroscopy at a single momentum transfer k/kF = 4.8 in our
group and has been published in previous work [80,189]. However, the normalisation
technique relied on experimental parameters and not the f-sum rule, and hence S(k)
6.5 SUMMARY 121
was less accurate. Both the f-sum rule and the analytic expression, equation 6.5,
provide a precise prescription to determine the contact for various 1/(kFa). The
transformation to values of the contact is as follows
INkF
= S↑↓ (k) · 4 ·(kFk
)−1 [1− 4
π
1
kFa
(kFk
)]−1
, (6.11)
where k/kF = 4.8. The experimental sequence is similar to the description in sec-
tion 6.3.2. After evaporation in a single optical dipole trap the magnetic field is
ramped to different values near the Feshbach resonance. The subsequent Bragg
pulse applied to the trapped gas was 50 µs long. The Bragg beams have been set up
in such a way that the momentum transfer results in a recoil frequency of ωR = 2π
300·103 s−1. The release time after the trap is switched off is 3 ms, and hence the
centre of mass displacement of the atomic distribution is chosen to yield the Bragg
spectra. The evaluation of the first moment of the line profiles follows from sec-
tion 6.3.2. Processing the Bragg spectra with equation 6.11 leads to a plot of the
interaction dependent contact, as shown in figure 6.7a. The transition from molecu-
lar excitations below the Feshbach resonance to atomic excitations above resonance
is smooth. As Tan predicted, the contact accurately connects the BEC and BCS
limit smoothly. As mentioned before, the theoretical challenges in describing the
BEC-BCS crossover arise from its strongly correlated nature and there is no small
parameter to set the accuracy of the theories. Techniques such as Monte-Carlo or
strong coupling theories lead to a number of quantitative properties, such as the
equation of state, frequency of collective oscillations, pairing gap, or the superfluid
transition temperature. However, other fundamental properties such as the single-
particle spectral function measured by rf-spectroscopy and the dynamic structure
factor probed by Bragg spectroscopy are not as well understood. The dynamic struc-
ture factors from Bragg spectra like those shown in figure 6.7a have been compared
to calculations based on a random-phase approximation and good agreement has
been found [192].
6.5 Summary
The structure factor of a strongly interacting ultracold Fermi gas follows a universal
law which is a direct consequence of Tan’s relation for the pair correlation functions.
The Bragg spectra for this experimental proof result from the centre of mass dis-
placement a cloud experiences following a Bragg pulse. The pair response and single
atom response are clearly distinguishable. Integration and the application of the f-
122 CHAPTER 6. UNIVERSAL BEHAVIOUR OF PAIR CORRELATIONS
−1010
5
10
15
20
1/(kFa)
I/(N
kF)
theoryexperiment
(a)
0 0.5 1 1.50
10
20
30
40
50
60
Cen
tre
of m
ass
disp
lace
men
t [µm
]Bragg frequency (ω / ω
R)
(b)
Figure 6.7: Interaction dependence of the contact in the BEC-BCS crossover ob-tained from Bragg spectra and equation 6.11. Inset: Bragg spectra representingS(k, ω) and showing how pair-correlations contribute to the spectral response [80].
sum rule lead to the static structure factor which represents a macroscopic measure
of the pair correlations. The contact in the BEC-BCS crossover is extracted from the
static structure factor using the universal law. These measurements provide one of
the first demonstrations of a broadly applicable exact result for strongly interacting
Fermi gases in the BEC-BCS crossover.
Chapter 7
Temperature dependence of the
contact
The strong interactions in the unitarity regime set the intrinsic properties of the
Fermi gas. Pair correlations persist up to temperatures higher than the critical
temperature, approaching TF , and hence raising the question of preformed pairs.
Chapter 6 highlighted the connection between the static structure factor and the
contact. In this chapter, measurements of the static structure factor are restricted
to the unitarity limit, where the contact is linearly proportional to the static struc-
ture factor. Therefore, a measurement of the static structure factor at different
temperatures gives direct information on the temperature dependence of the con-
tact.
7.1 Introduction
The contact depends on the interaction strength 1/kFa and the temperature T/TF
of the Fermi gas. The concept of the contact was introduced in chapter 2. In this
chapter, the goal is to measure the contact in a unitary Fermi gas for increasing
temperatures. At unitarity, a → ∞, the second term in brackets in equation 6.5
vanishes and the spin anti-parallel static structure factor reduces to
S↑↓ (k ≫ kF ) =I
NkF
kF4k. (7.1)
The static structure factor now becomes directly proportional to the contact. If the
temperature is increased, the two-body correlation function should decrease, as more
single particle states become occupied. Hence the contact is predicted to decrease
with increasing temperature.
123
124 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
Different theoretical models have been developed to calculate the temperature
dependent contact, in the homogeneous as well as the trapped case. In the high
temperature limit for a trapped gas, T > 0.5 TF , the fugacity is small and the virial
quantum cluster expansion is accurate [135]. In the low temperature regime around
and lower than the critical temperature TC , various many-body t-matrix approxima-
tion based models are used [84,85,149,165]. Also, quantumMonte-Carlo calculations
have been performed [75,131,132]. All the predictions differ slightly near the critical
temperature as the low and high temperature regimes have been connected differ-
ently. The contact at T = 0 seems to lie in the range 3 − 3.4 I/(NkF ). A precise
measurement of the contact therefore would resolve the discrepancy between the dif-
ferent approaches and help to decide which underlying model is the most descriptive
for the unitarity regime.
At unitarity, one can distinguish two temperature thresholds, the onset of
pairing at T ∗ and superfluidity at TC . The question arises, how the contact is linked
to the pairing temperature [239], as pair-correlations persist up to temperatures
higher than TC . While this thesis was being written, a debate about the nature of
a strongly interacting Fermi gas above TC has been going on, i.e. whether it is a
Fermi liquid [173] or a state with an excitation gap/pseudogap [86, 240, 241]. Pair-
correlations above TC have been seen in imbalanced Fermi gases with radio-frequency
spectroscopy [41,60,220] and pairing without superfluidity has been observed.
In solid state physics, evidence of pairing above TC has been demonstrated
for cuprate superconductors through dispersion measurements close to the Fermi
surface [242]. These type of experiments lack the possibility to directly measure
the (momentum) pair correlation functions. However, they can search for features
unique to Cooper pairing, e.g. features which are present in the electronic excitation
spectrum above TC .
In this chapter, two experiments are presented to obtain Bragg spectra and
dynamic structure factors, either with distributions imaged immediately after the
Bragg pulse, S∆P (k, ω), or after equilibration, S∆E(k, ω). The outline of the exper-
imental sequence is the subject of section 7.2. The principal ideas and differences of
the experiments is discussed in section 7.3. Bragg spectra that yield S∆P (k, ω) from
the centre of mass displacement as well as from the width of the atomic distribution
are explained in detail in section 7.4. The acquisition of S∆E(k, ω) follows in sec-
tion 7.5. From the dynamic structure factors the values of the static structure factor
and contact are determined. The results are discussed in section 7.6 together with
theoretical predictions. Note that for the remainder of this thesis the momentum is
labelled in terms of the wave vector, i.e. q/~ = k, where |k| = k, which is different
7.2 EXPERIMENTAL VERIFICATION 125
Figure 7.1: Optical Bragg beam setup for temperature dependence measurements.On the left is a top view and on the right is a side view. Note in the side view thatthe Bragg beams have a different angle to the trap laser.
from the notation in chapter 3.
7.2 Experimental verification
The experimental verification of the temperature dependence of the contact is per-
formed in a different setup from subsection 6.2 and is sketched in figure 7.1. After
evaporation the atoms are adiabatically transferred in 600 ms into the second sin-
gle far-detuned dipole trap where N↑ = 180 000 particles are measured in one spin
state at (0.09 ± 0.05) T/TF . The combination of the residual magnetic field curva-
ture and optical trap yield homogeneous trap frequencies of ωz = 2π 210 s−1, ωx =
2π 56 s−1, ωy = 2π 24.5 s−1, or ω = 2π 66 s−1; hence, the Fermi energy defined by
ϵF = ~ω (6N↑)1/3 is ϵF = 2π 7.2·103 s−1 and the Fermi wave vector is kF =
√2mϵF/~
= 2.9 µm−1. For more details on the trap frequencies see section 4.3.3.
This is the starting point for every heating sequence as it is intended to main-
tain the same Fermi energy for all experiments. The confining trap is switched off for
a variable time before non-adiabatically ramping it back on [48]. Rethermalisation
is allowed for τtherm = 400 ms which is larger than 10 ω−1y . The trap allows heating
126 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
to a maximum temperature of 1.0 T/TF without a significant loss of atoms. The
determination of the temperature is described in chapter 5. The temperatures are
obtained for each Bragg spectrum individually and the release time for absorption
imaging is usually τTOF = 2 ms. The relative uncertainty in temperature is about
7 %.
The Bragg setup is designed so that two low-power laser beams of PBr = 40 µW
intersect at an angle θBr = 49.5 to form a moving lattice potential at the position of
the atoms. The velocity of the potential is determined by a small tunable frequency
difference ω. In this configuration, the Bragg resonance condition leads to a recoil
energy ωR = 2~k2/m = 2π 51.6·103 s−1 and a pair frequency ωpair = ωR/2 − 2π
25.8·103 s−1. The momentum transfer to the atoms is given by the ratio k/kF =
2.7 and is characterised by the recoil momentum of ~k = 2~kL sin(θ/2), kL being
the laser wave vector. The Bragg pulse is applied for τBr = 200 µs on the trapped
atoms to obtain a Bragg spectrum. The resulting Fourier width is ≤ 2π 800 s−1,
which is small compared to ωR and therefore is ignored from now on.
7.3 Momentum and energy interrelation
Bragg spectra so far are based on the traditional centre of mass displacement,
∆X(k, ω), associated with the relative momentum transfer, ∆P (k, ω), for a particle
or pair with mass m after a time ∆t, i.e. ∆P (k, ω) = m∆X(k, ω)/∆t. To measure
the transferred momentum the trap is switched off shortly after the Bragg pulse and
after a short release time the cloud distribution is imaged. Therefore, the dynamic
structure factor is linked to the momentum such that ∆P (k, ω) = Nsc~k ∝ S (k, ω)
for Nsc scattered particles, and hence linked to ∆X(k, ω). The notation, ∆P , refers
to the way the Bragg spectrum is obtained, i.e. after a short release time. Note that
the Bragg pulse transfers always momentum ~k and energy ~ω.However, the dynamic structure factor can also be extracted from the rela-
tive energy transfer, ∆E(k, ω). The energy of the transferred momentum is man-
ifested by an increase of the width, ⟨σ2(k, ω)⟩, such that ∆E(k, ω) = ~ω · Nsc =
(m/2)ν2∆ ⟨σ2(k, ω)⟩ ∝ ωS(k, ω). The last step shows that the energy is propor-
tional to the first moment of the dynamic structure factor with respect to energy.
Both ∆P (k, ω) and ∆E(k, ω) are proportional to the number of scattered
photons, Nsc. The momentum depends on the zeroth moment of S(k, ω) while the
energy represents the first moment of S(k, ω) [197]. The momentum and energy are
7.3 MOMENTUM AND ENERGY INTERRELATION 127
Spectrum Equilibration Profile Quantity Relation to S(k, ω)∆Pim(k, ω) 0 asymmetric ∆X k S(k, ω)∆Eim(k, ω) 0 asymmetric ∆σ2
x,∆σ2y ω S(k, ω)
∆Eeq(k, ω) 400 ms symmetric ∆σ2x,∆σ
2y ω S(k, ω)
Table 7.1: Summary of the evaluation procedures. Images taken shortly after theBragg pulse are indicated by ‘0’. ∆X denotes the centre of mass and ∆σ2
x,∆σ2y the
width in the x- or y-direction, respectively.
related to the S(k, ω) according to equation 3.45 by
S (k, ω) ∝ ∆P (k, ω)
k=
∆E (k, ω)
ω. (7.2)
Two different experiments are performed in the following. On the one hand, the
clouds are imaged immediately after the Bragg pulse and the respective spectra
carry the index ‘im’. On the other, the cloud is held in the trap to equilibrate and
any centre of mass movement is damped out. These Bragg spectra are indexed with
‘eq’. First, the conventional method is applied which leads to asymmetric atomic
distributions. In section 7.4.1 it is explained how the centre of mass displacement,
∆X(k, ω), along the Bragg scattering direction is evaluated from the images. Nor-
malised Bragg spectra are labelled as ∆Pim(k, ω). Second, the same experimental
data set is used in section 7.4.2 to extract the width, ∆σ2. Normalised Bragg spec-
tra are denoted by ∆Eim(k, ω). Note that the evaluation of the second moment
of a line profile coincides with the expression of the energy transfer, despite the
distribution not being in thermal equilibrium. Third, in section 7.5, another exper-
iment is performed with equilibrated, and hence symmetric, atomic distributions.
The evaluation of the widths provide Bragg spectra labelled as ∆Eeq(k, ω). The
method of equilibrated atoms in the trap should examine if an alternative approach
to the evaluation of the centre of mass displacement provides more accurate results.
This might be important when other experiments are performed, for instance, at
low momentum transfer (k < kF ). The momentum transfer with a short release
time and the energy transfer with an equilibration time have previously been used
in optical lattice experiments with bosons [185–187]. Table 7.1 summarises briefly
the strategies for the upcoming sections.
128 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
7.4 Bragg spectra after immediate release
The momentum transfer method has been applied in previous work [80]. A graphic
outline is shown figure 6.3. The slight modifications in the process from single pic-
tures to the final dynamic structure factor is described in the following. After the
cloud is heated a Bragg pulse is applied while the dipole trap is on and the total time
between the Bragg and imaging pulse is always 3 ms. For temperatures below 0.4
T/TF , the cloud travels after the Bragg pulse further in the trap for 1 ms. Then the
trap is switched off and the atoms are released for 2 ms before an absorption image is
taken. When the temperature is above 0.4 T/TF , the atoms are held after the Bragg
pulse another 2.95 ms in the trap and the time-of-flight before imaging is reduced
to 0.05 ms. The reason for this separation is to aim for a good signal-to-noise in
the images. The hold time in the trap is always short compared to the inverse trap
frequencies. So the hold time does not influence the measured displacements. As
seen later, the results are well maintained by the f-sum rule in the normalisation,
which bypasses technical details such as release times.
From previous measurements in this group, the centre of mass displacement
turned out to be a robust method [80], as elastic collisions after the applied Bragg
pulse preserve the centre of mass motion, ∆X, throughout the crossover regime.
Furthermore, ∆X is less sensitive to atom number fluctuations. Particle conser-
vation requires a summation over all involved scattered atoms, and therefore it is
important to count all atoms but to avoid the noise of pixel counts. In the former
experiments [80], a momentum transfer of k = 5 kF was used. For that momentum,
scattered atoms eventually separate from the parent cloud leaving a gap in-between.
In a summation over the region of interest, this gap may added noise. In the present
work, the momentum transfer is reduced to k = 2.7 kF and thus the centre of mass
displacement is less enhanced for the same time-of-flight after the Bragg pulse. In-
stead of a separate cloud, the distribution becomes very distorted but contiguous
and the region of interest includes less noise from pixels without signal.
Another advantage of a lower momentum is an increase of the contribution of
the spin anti-parallel structure factor, according to equation 7.1, since the relative
momentum determines the probed pair size. The lower k/kF , the larger the probed
pairs can be. In general, the signal of atoms at the edges of the distorted cloud
might become as small as the noise of the pixel counts. The region of interest has
to be selected carefully which means that in the direction of integration it has to be
as small as possible to include all of the atoms but to exclude background that only
adds noise.
7.4 BRAGG SPECTRA AFTER IMMEDIATE RELEASE 129
For the evaluation of a single image it is assumed that the pictures have a hori-
zontal axis labelled as x which is also the direction of the applied Bragg momentum.
The vertical, perpendicular direction to the Bragg scattering in the image is labelled
with y. Each image i is taken for a different Bragg frequency ω. Compared with re-
sults in the previous chapter, an additional fringe removal procedure, see section 4.5,
is applied with respect to the reference pictures to reduce uncorrelated noise.
7.4.1 Bragg spectra from the first moment
Building Bragg spectra
For every image i, a one-dimensional sum is taken in y-direction to obtain a line
profile in the x-direction, ni(x). Each line profile ni(x) is normalised to unity. The
first moment of ni(x) is equivalent to the centre of mass and labelled ‘(1)’. Hence,
the centre of mass follows as
X(1)i =
∑x ni(x) · x∑x ni(x)
. (7.3)
As mentioned above, the cloud distribution is asymmetric after a short release time
and a gaussian fit to approximate the line profile in order to gain a better signal-to-
noise cannot be applied as this could omit part of the signal.
To quantify the centre of mass displacement, ∆X(1)i , for each ni(x), the position
of the unperturbed cloud, X(1)off , needs to be known. It is convenient to use values
from off-resonant Bragg frequencies and subtract X(1)off from ∆X
(1)i . Error may occur
when the trap centre drifts during one sequence, e.g. due to thermal lensing at the
spot where the laser enters the glass cell or other optical elements. The picture
number is related to the Bragg frequency, so therefore i is now replaced by the
Bragg frequency i→ ω, and so X(1)i ≡ X(1)(ω). Thus a Bragg spectrum, where the
centre of mass of ni(x) has been evaluated, may be written as
∆X(1) (ω) = X(1) (ω)−X(1)off . (7.4)
Derivation of the dynamic structure factor
As shown in section 3.8, the centre of mass displacement spectrum, ∆X(ω), is pro-
portional to the dynamic structure factor. In order to calculate the static structure
factor, the Bragg spectra need to be normalised and the f-sum rule has to be applied,
see section 3.8.4. In the previous experiments, the signal of the centre of mass was
130 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
ω/ωR
∆Pim
(k,
ω/ω
R)
T/T
F
0.090.170.270.550.840.96
Figure 7.2: Bragg spectra from the centre of mass displacement, ∆Pim(k, ω), aftermomentum transfer for different temperatures T/TF . At high temperatures thespectrum is dominated by uncorrelated atoms gaussian-distributed around the recoilfrequency ωR. The pair signal evolves at the pair frequency (0.5 ωR) for decreasingtemperatures.
easily distinguishable due to higher momenta and low temperatures, and therefore
a model independent summation over the pixel was sufficient. The recoil energy
was higher (ωR = 2π 295·103 s−1) and the onset of the spectrum was clearly around
2π ·103 s−1. Now, the probe momentum is lower (k = 2.7 kF ) and the resonance
frequency is 6 times lower (ωR = 2π 51 ·103 s−1), which means that a spectrum has
significantly non-zero values at very low frequencies. These values are even more
pronounced for increasing temperatures. When the f-sum rule is applied, small fluc-
tuations of S(k, ω) in the low frequency range will cause large errors. Moreover, the
Bragg spectra are weighted for increasing temperatures by the Boltzmann factor and
the denominator in equation 3.47 deviates from unity. The dynamic structure factor
for higher temperatures contains then a significant weight at negative frequencies
where it is impossible to obtain data. This necessitates fitting to the Bragg spectra,
which are to a first approximation two gaussians.
First, the Bragg spectra need to normalised to the atom number, which is
governed by the f-sum rule, NωR = ~∑
ω ωS(ω). So the normalised spectrum
∆Pim(ω) follows as
∆Pim (ω) =∆X(1) (ω)∑
ω ∆X(1)(ω) · ω
. (7.5)
7.4 BRAGG SPECTRA AFTER IMMEDIATE RELEASE 131
The fit function consists of two gaussians multiplied by the detailed balancing term;
hence
∆Pim,fit(ω) =
[a1e
− (x−c1)2
2b21 + a2e− (x−c2)
2
2b22
]·(1− e−~ω/kBT
). (7.6)
The centre positions of the gaussians are fixed to the excitation frequencies c1 =
ωR/2, c2 = ωR but the amplitudes a1, a2 and widths b1, b2 are free. The parameter
T/TF is known and ~ω/kBT = (ω/ωr) (T/TF ) is defined in terms of the recoil
frequency ωR. A gaussian may not describe the coldest spectra well. The estimated
error of ∆Pim,fit(ω) to the data is about 4% for the coldest cloud, 1% at 0.5 T/TF
and decreasing for even higher temperatures. For this estimation, a calculated zero
temperature spectrum at 3 k/kF is taken [235] and the same fit routine is applied
as for the spectra.
The real dynamic structure factor, Sim,∆P (k, ω), after the immediate release
time is for any momentum k
Sim,∆P (k, ω) =∆Pim,fit(k, ω)
1− e−~ω/kBT(7.7)
These Bragg spectra are shown in figure 7.2. The solid lines connect the data points
and the dashed line represents the fits from which S(k, ω) is extracted. At low
temperatures the pair peak near ω = 0.5 ωR in the spectrum is well pronounced. For
increasing temperatures T/TF , the pair peak smoothly vanishes and the spectrum
is more and more dominated by response at the atomic resonance.
7.4.2 Bragg spectra from the second moment
Building Bragg spectra
The same line profiles ni(x) as used in the previous section can also provide the
second moment, which is equivalent to the mean square size and thus the energy of
the cloud distribution. The second moment, labelled ‘(2)’, of the asymmetric line
profiles may be written as
X(2)i =
∑x ni(x) · (x−X0,COM)2∑
x ni(x)(7.8)
whereX0,COM is the centre of mass of an unperturbed cloud and is the same for every
image i. When looking at the second moment, there is also information stored in
the direction perpendicular to the Bragg pulse, ni(y), where the cloud is symmetric.
132 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
For every image i, a sum can be taken in the x-direction to obtain a line profile
in the y-direction, ni(y). After summation the area under each line profile ni(y) is
normalised to unity. While the ni(x) are asymmetric, the ni(y) are symmetric. A
priori there is no reason to fit a function to ni(y), or especially to expect a gaussian
to work. However, a gaussian fit to the symmetric line profile in the y-direction
seems to be reliable compared to an evaluation of the second moment from the raw
data due to a gain of better signal-to-noise. One may use a gaussian function, such
as
g (x) = a · e−(x−c)2
2b2 , (7.9)
where a is the amplitude, c the centre position and b the width. During the pro-
cessing it has been easier to look at the second moment to check the sensitivity
of the fit; thus second moment of the fit function is g (x) · (x− c)2. Equation 7.8
has been used to calculate the second moment in the y-direction and similarly one
obtains Y(2)i . At the end, two spectra are obtained from the second moment of the
line profiles: X(2)i based on the raw data and no fit and Y
(2)i based on a fit to the
second moment of each line profile. The spectra X(2)i and Y
(2)i are equal to the
mean square size, and thus X(2)i ≡ σ2
x(ω) and Y(2)i ≡ σ2
y(ω). The interest lies in the
relative increase in width with the applied Bragg frequencies. Therefore, the widths
of the unperturbed cloud, X2off and Y 2
off , have to be determined. Conveniently, one
can use the width values where an off-resonant Bragg pulse was applied. Then, the
Bragg spectra following from the mean square sizes of the line profiles are
∆X(2) (ω) = X(2) (ω)−X(2)off ≡ ∆σ2
x (ω) , (7.10)
∆Y (2) (ω) = Y (2) (ω)− Y(2)off ≡ ∆σ2
y (ω) . (7.11)
These Bragg spectra are plotted in figure 7.3.
Derivation of the dynamic structure factor
The normalisation of the Bragg spectra from the widths is set to unity, because the
second moment is proportional to the term which appears in the f-sum rule integral.
Thus
∆Eim,x (ω) =∆σ2
x (ω)∑ω ∆σ
2x (ω)
. (7.12)
7.4 BRAGG SPECTRA AFTER IMMEDIATE RELEASE 133
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
ω/ωR
∆Eim
(k,
ω/ω
R)
T/T
F
0.090.170.270.550.840.96
Figure 7.3: Bragg spectra from the mean square size, ∆Eim(k, ω), after immediaterelease for different temperatures T/TF . At high temperatures the spectrum isdominated by uncorrelated atoms gaussian-distributed around the recoil frequencyωR, which is weighted by ω as the energy depends on the first moment of Sim(k, ω).The pair signal evolves at the pair frequency (0.5 ωR) multiplied by the weightingω for decreasing temperatures.
The same expression yields ∆Eim,y (ω). Both can be averaged to
∆Eim (ω) =∆Eim,x (ω) + ∆Eim,y (ω)
2. (7.13)
A function with two gaussians times the detailed balancing term and frequency is
fit to ∆Eim (ω), where the fit function is similar to equation 7.6 multiplied by the
frequency ω; hence
∆Eim,fit (ω) = ω ·
[a1e
− (x−c1)2
2b21 + a2e− (x−c2)
2
2b22
]·(1− e−~ω/kBT
). (7.14)
Again, the centre positions of the gaussians are fixed to c1 = ωR/2 and c2 = ωR,
respectively, but the amplitudes a1, a2 and widths b1, b2 are free. The real dynamic
structure factor Sim,∆E (k, ω) is calculated using equation 7.7, but with a weighting
of ω
Sim,∆E (ω) =∆Eim,fit (ω)
ω (1− e−~ω/kBT )(7.15)
134 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
The two results from the first and second moment, Sim,∆P (k, ω) and Sim,∆E(k, ω),
lead to the same S(k, ω) and can be averaged. However, the noise in S(k, ω) obtained
from the second moment is large at low frequencies. The reason is the division
by the frequency in equation 7.15 and for low frequencies the noise becomes large.
To compensate for this artefact a weighting of Sim,∆P (k, ω) and Sim,∆E(k, ω) is
considered. The weighting is chosen such that the contributions from Sim,∆P (k, ω)
and Sim,∆E(k, ω) are equal at high frequencies (ωmax) but at low frequencies the
contribution of Sim,∆E (k, ω) is small
Sim (k, ω) = Sim,∆P (k, ω)
(1− 1
2
ω
ωmax
)+ Sim,∆E (k, ω)
(1
2
ω
ωmax
). (7.16)
The equal weighting at high frequencies is possible due to equation 3.45 which states
that the momentum transfer and energy transfer contribute by the same factor to
S(k, ω).
−0.5 0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
ω/ωR
S im(k
,ω/ω
R)
T/T
F
0.090.170.270.550.840.96
Figure 7.4: Dynamic structure factor, Sim(k, ω), after averaging the dynamic struc-ture factors of the Bragg spectra from the first (centre of mass displacement) andsecond moment (width). Atoms respond at ω = ωR while pairs contribute at 0.5ωR. Note that Sim(k, ω) has non-zero values at negative frequencies.
Figure 7.4 presents Sim (k) from the momentum and energy transfer of Bragg
spectra with immediate imaging after the Bragg pulse. The static structure factor
7.5 BRAGG SPECTRA AFTER EQUILIBRATION 135
Sim (k) is determined from equation 3.55,
Sim (k) =
∑ω Sim (k, ω)∑
ω Sim (k, ω) · ω· ωR. (7.17)
The result is discussed in section 7.6.
In section 3.7 the fluctuation dissipation theorem was considered. At the end,
it was shown that instead of S(k, ω) at higher temperatures, the dissipative part of
the correlation function, χ′′, can be used. The difference is a coth-function, which
diverges at zero energy and shows an upward tail. When comparing χ′′ and S
over the applied temperature range, this signature is marginal. The noise in the
spectra at these frequencies is larger and cannot be compared with the artefact of
the coth-function.
7.5 Bragg spectra after equilibration
So far, it has been tested to see if the evaluation of the second moment of the
line profiles for non-equilibrated atomic distributions gives the same result as the
traditional evaluation of the centre of mass. Another approach to measure the energy
transferred is to allow the cloud time in the trap to equilibrate after the Bragg pulse
again. The atoms are held after the Bragg pulse in the trap which is designed to be
deep and therefore holds atoms at higher temperatures. The question arises whether
performing the evaluation on an equilibrated cloud provides better signal-to-noise
and whether Bragg scattering experiments for k < 2 kF (where the signal from the
centre of mass is small) still shows well pronounced features of a Bragg spectrum.
In order to allow the atoms to equilibrate after the Bragg pulse, the trap is kept
on for another 400 ms after the Bragg pulse. The deep trap allows no loss of atoms
following the Bragg pulse up to temperatures of 1.0 T/TF . The equilibration damps
all centre of mass movement and the signal will be manifested as a pure increase in
width of the cloud size. In-situ images are taken for temperatures T/TF ≥ 0.4 and,
with a time of flight, when T/TF < 0.4. Because the atomic distributions for each
Bragg frequency are symmetric, the second moment can be extracted in both the
directions parallel and perpendicular to the Bragg pulse direction, which are the x-
and y-directions.
Building Bragg spectra
For every image i, a one-dimensional sum is taken in the y- x-direction to obtain
symmetric line profiles in the x- y-direction, ni(x) ni(y). In both directions,
136 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
a gaussian function as in equation 7.9 can be fitted, where for evaluation purposes
the second moment, g (x) · (x− c)2, is monitored. Applying this expression reduces
the in-profile pixel noise and the spectra are only dominated by the shot-to-shot
uncertainty of the atom number. An example of the single gaussian fit to a line
profile is shown in figure 7.5. The area under each line profile ni(x) ni(y) is
0 50 1000
0.01
0.02
0.03
a.u.
0 50 1000
2
4
pixel
a.u.
10 20 30 40
0
10
20
30
40
50
60
image i
a.u.
Figure 7.5: Example of fitting a single line profile of an image. On the left is shownthe integrated line profile (top) and its second moment (bottom). The gaussian fit(blue) to the data (black points) improves the signal-to-noise. On the right the totalspectrum is shown with (red) and without the improvement (black points) due tothe fit. Black points correspond to individual pictures i.
normalised to unity. The second moment is labelled ‘(2)’, as before, and represents
again the mean square size. It may be written as
X(2)i =
∑x ni(x) · (x−X0)
2∑x ni(x)
(7.18)
Y(2)i =
∑y ni(y) · (y − Y0)
2∑y ni(y)
(7.19)
7.5 BRAGG SPECTRA AFTER EQUILIBRATION 137
where X0 Y0 is the centre of mass of an the cloud and is now calculated for every
image i. This approach is insensitive to shot-to-shot drifts in the centre of mass
position of the cloud. At the end, two spectra are obtained, X(2)i,fit and Y
(2)i,fit for the
mean square size in both the x- and y-directions. In order to work out the energy
transferred to the cloud, one needs to compare the widths after a Bragg pulse with
the widths of an unperturbed cloud, X(2)fit,off and Y
(2)fit,off . These can be taken from
values where a Bragg pulse has a far off-resonant frequency was applied. Replacing
the picture number with the Bragg frequency i→ ω relates the spectra to the mean
square size X(2)i,fit ≡ σ2
x(ω) and Y(2)i,fit ≡ σ2
y(ω). The spectra are now
∆X(2)i,fit (ω) = X
(2)i,fit (ω)−X
(2)fit,off ≡ ∆σ2
x (ω) (7.20)
∆Y(2)i,fit (ω) = Y
(2)i,fit (ω)− Y
(2)fit,off ≡ ∆σ2
y (ω) . (7.21)
These spectra from the equilibrated cloud are normalised for both directions accord-
ing to equations 7.12
∆Eeq,x (ω) =σ2x (ω)∑
ω σ2x (ω)
(7.22)
and averaged according to equation 7.13,
∆Eeq (ω) =∆Eeq,x (ω) + ∆Eeq,y (ω)
2. (7.23)
These spectra from equilibrated clouds are depicted in figure 7.6 and resemble the
spectra of figure 7.3 but with significantly less noise.
Derivation of the dynamic structure factor
The spectra are fitted again with a function of two gaussians, as in equation 7.14,
to obtain the actual dynamic structure factor like equation 7.15
Seq (k, ω) = Seq,∆E (k, ω) =∆Eeq,fit (ω)
ω (1− e−~ω/kBT ), (7.24)
for any momentum k. Seq (k, ω) is plotted for some different temperatures in fig-
ure 7.7. The static structure factor is calculated also from equation 3.55; thus
Seq (k) =
∑ω Seq (k, ω)∑
ω Seq (k, ω) · ω· ωR. (7.25)
138 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
ω/ωR
∆Eeq
(k,
ω/ω
R)
T/T
F
0.090.170.280.550.860.97
Figure 7.6: Bragg spectra from the mean square size, ∆Eeq(k, ω), after energytransfer for different temperatures T/TF . At high temperatures the spectrum isdominated by uncorrelated atoms gaussian-distributed around the recoil frequencyωR. Again, the spectra a weighted by ω due to the evaluation of the relative energytransfer. The pair signal evolves at the weighted pair frequency (0.5 ωR · ω) fordecreasing temperatures.
−0.5 0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
ω/ωR
S eq(k
,ω/ω
R)
T/T
F
0.090.170.280.550.860.97
Figure 7.7: Dynamic structure factor, Seq(k, ω), after energy transfer for differenttemperatures T/TF . At high temperatures the spectrum is dominated by uncorre-lated atoms gaussian-distributed around the recoil frequency ωR. The pair signalevolves at the pair frequency for decreasing temperatures. Due to the division bythe frequency to obtain Seq(k, ω) from the energy, the noise is pronounced at lowfrequencies.
7.6 TEMPERATURE DEPENDENCE OF THE CONTACT 139
Even though these dynamic structure factors appear slightly noisier than those in
figure 7.4, these are much less noisy than what would be obtained from the data of
figure 7.3 alone.
7.6 Temperature dependence of the contact
Once the static structure factors, Sim(k) and Seq(k), are determined, they are trans-
lated into the contact I/(NkF ) using equation 7.1. The contact for different tem-
peratures is plotted in figure 7.8 for values from the momentum transfer spectra
(triangles) and from the energy transfer spectra (circles). The contact decreases
smoothly from ∼ 3.0 to 0.5 for increasing temperatures T/TF = 0.1 − 1.0 and a
discontinuity in the critical temperature region is not visible. The data has vertical
errors due to atom number fluctuations and statistical uncertainties resulting from
the number of spectra used. A change in the Fermi energy, thus k/kF , due to atom
number fluctuations is marginal (maximum 4%). An upper bound of the vertical
error can be given by the experimental limit of extracting the static structure factor,
i.e. the S(k) is taken from momentum transfer Bragg spectra of a spin polarised
gas. The temperature measurement has been discussed in chapter 5.
The contact at the lowest temperature is I/(NkF ) = 3.1± 0.3, which is slightly
lower but still in agreement with I/(NkF ) = 3.4 ± 0.2 taken from the equation of
state [173]. Note that the contact values are clearly non-zero for T > TC indicating
a gradual build up of pair-correlations. This is not to be confused with pseudogap
effects, as the probe momentum in this experiment is 2.7 k/kF , which probes the
high momentum tail of the momentum distribution. Therefore, the signal seen here
is dominated by the 1/k4 tail and not a pseudogap pairing, which would occur near
the Fermi surface k/kF ≈ 1 [240]. Note that the potential of the contact to explain
the pairing gap is unclear due to the definition in the high momentum limit while
the structure factor might still be suitable to explain the pairing gap due to the link
to the pair-correlation function between opposite spins. Therefore, equation 7.1 has
to be given careful consideration for transferred momenta of k/kF → 1.
The above experimental result is compared to the theoretical models for the
trapped Fermi gas based on many-body t-matrix approximations, which are the
gaussian pair fluctuation (GPF) [135], self-consistent approach (GG) [85] and non-
self-consistent approach (G0G0) [84]. The high temperature region uses the second
(b2) and third (b3) order of the accurate quantum virial expansion method [135].
While all theories agree with a decay of the contact for temperatures larger
than TC , they deviate in the low temperature regime around and below TC . The
140 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
3.5
T/TF
I/(N
k F)
G
0G
0 (Palestini et al.)
GG (Enss et al.)GPF (Hu et al.)Virial 2Virial 3Bragg (eq)Bragg (im)
Figure 7.8: Temperature dependence of the contact. The contact at zero tem-perature lies between I = 3 and 3.4 and decays monotonically to values ≤ 0.5 forT/TF = 0.1 − 1.0. The data points are obtained using equation 7.1 for the staticstructure factor of Bragg spectra after momentum transfer (triangles) and energytransfer (circles). Vertical error bars are due to statistical errors of eight spectraand atom number fluctuations. The horizontal errors reflect the uncertainties in thetemperature determination. The solid lines are the theoretical predictions based ont-matrix approximations (GPF, GG, G0G0) and the dashed and dashed-dotted linesresult from a virial expansion calculation.
7.6 TEMPERATURE DEPENDENCE OF THE CONTACT 141
differences are even more visible in the calculation for the homogeneous case. There-
fore, the homogeneous case is briefly revisited: The G0G0 theory predicts a max-
imum around the critical temperature TC due to a strengthening of local pairing
correlations in the absence of long-range order. Then, quasi-particle (phonons etc.)
correlations may contribute to the momentum distribution. The G0G0 theory also
assumes that the phonon mode is dominant in the whole superfluid phase. The same
assumption is in disagreement with the GPF model which states that the phonon
mode is effective only in a very narrow window (T ≤ 0.07 TF ≪ TC) where the
contribution from phonons is of relative order 10−3. Another calculation uses the
assumption of a Fermi liquid behaviour for a weakly coupled Fermi gas [165] and
suggests that phonon excitations will enhance the contact according to a temper-
ature behaviour ∝ T 4 and maximises at around 0.5 T/TF . However, the Fermi
liquid theory actually strongly overestimates the contact at TC [135]. In contrast,
the calculations with the GPF and GG approach find that the contact decreases
monotonically as the temperature increases, except at T ≪ TC , where the contact
plateaus. Near the critical point strong pair fluctuations occur. The strong coupling
theories cannot handle these divergences properly. Therefore, it is not clear if the
peak in the G0G0 theory is due to an increase of the contact or due to a breakdown
of the approximations in this approach. So far, the literature has not reported on
the temperature dependent contact calculated with the renormalisation group the-
ory to account for the criticality. Note that in the weakly interacting limit already
at 1/(kFa) = − 1 all strong coupling theories for the contact are in qualitative
agreement [135]. Since the contact is predicted to decrease with temperature in
the superfluid phase it is most likely to find the same decrease for a unitary Fermi
superfluid.
The same theories predict concordantly for the homogeneous case that C de-
cays slower at high temperatures than in the trapped case. This weak temperature
dependence is because the contact is not defined for long-range order, see equa-
tion 2.7. However, quasi-particles are probed in the low-momentum limit, for which
the contact is partly defined. A more precise measurement in the homogeneous case
would be necessary to make a statement about the detailed predictions.
In the zero temperature limit, it is possible to calculate the contact density ζ
analytically at unitarity from C/(nkF ) ≡ I/(NkF ) = 512ζ/(175ξ1/4) where ξ is given
by the universal parameter [65, 135]. According to this, the contact in the trapped
case scales linearly with the contact density but the decay at higher temperatures
is non-trivial. For the theoretical values I/(NkF ) = 3.0 − 3.4, ζ = 0.92 − 0.82.
From this experiment, the averaged contact is about I/(NkF ) ≈ 2.85 for the coldest
142 CHAPTER 7. TEMPERATURE DEPENDENCE OF THE CONTACT
sample at 0.1 T/TF , which yields ζ = 0.79(7). The ENS result is ζ = 0.93(5) at
0.03(3) T/TF [130]. Finite temperatures seem to have an immediate effect, which
is consistent with the interaction dependent contact measurements [133,234], where
the values at 1/(kFa) = 0 seem to be systematically lower than expected.
Though the contact can be measured, more detailed theoretical as well ex-
perimental analysis is necessary as open questions remain. It is still not clear how
the contact evolves from the well defined short-range structure r ≪ (kF )−1 to the
medium-range limit r ∼= (kF )−1 [84], i.e. the details on the transition behaviour
from single-particle to pseudogap pairing and collective excitations.
7.7 Summary
The temperature dependence of Tan’s contact I/(NkF ) has been measured at uni-
tarity with inelastic two-photon Bragg spectroscopy. The evaluation is based on the
dynamic structure factor from traditional centre of mass displacement and energy
transfer measurement. Two sets of measurements have been performed to show that
both the energy of an immediately released as well as equilibrated cloud leads to the
dynamic structure factor. The resulting static structure factor yields the contact
from a universal law, see equation 7.1. The values for the contact, I/(NkF ), agreequalitatively with the theoretical predictions but the precision of the experimental
values is not good enough to resolve the differences between the theoretical models.
If the experimental accuracy can be improved, it would be interesting to focus on
the contact in the region around the critical temperature. From the fact that the
contact persists for temperatures higher than the critical temperature TC , it can be
concluded that the pair-correlations exist above the superfluid transition tempera-
ture. The dichotomy whether or not these relate to pairing near the Fermi surface
where pseudogap effects are expected remains an open question. The energy trans-
fer method of Bragg spectroscopy may give improved signal-to-noise in the phonon
regime, i.e. with a relative momentum transfer k/kF ≤ 1, to investigate thermo-
dynamics of quasi-particles. Access to pair-correlations at a macroscopic level may
give insight to the pseudogap regime found in high-TC superconductors [242].
Conclusion and Outlook
In conclusion, aspects of a recently proposed quantity, the universal contact, have
been investigated for dilute strongly interacting Fermi fluids with customised tech-
niques. The contact parameter is measured with Bragg spectroscopy for different
large momenta, k ≫ kF , as well as for different temperatures, in line with theoretical
predictions of several theory groups. This contact parameter will find application
in future work as it can quantify the temperature-coupling phase diagram and may
be used to help understand the nature of unitary Fermi gases.
Summary
The apparatus produces ultracold Fermi gases of 6Li by laser and evaporative cooling
in an ultra-high vacuum glass cell. At high magnetic fields around 834 G, the
interaction between different spin states can be varied from tightly bound molecules
to weakly correlated pairs, using the broad Feshbach resonance.
Universal static structure factor
On resonance, when the s-wave scattering length exceeds the interparticle spacing,
the gas is unitarity limited. Universal relations, first derived by Tan, accurately
describe the unitarity regime. Particularly in chapter 6 of this work, the universal
relation for the static structure factor, S(k), is verified above and below the Feshbach
resonance over a wide range of momenta using Bragg spectroscopy. The measured
Bragg spectra quantify the linear response to the density-density correlations and
lead to the dynamic and, consequently, the static structure factor. Sum rules enable
us to bypass details of some difficult experimental parameters and provide the correct
normalisation of the spectra. Since only the relative momentum distribution is
important to show the universal law, the Bragg momentum transfer is fixed by the
geometry of the Bragg beams and the Fermi momentum is tuned by varying the
geometric mean trapping frequencies of the crossed beam dipole trap. In contrast to
143
144 CONCLUSION AND OUTLOOK
previous work, the Bragg beams are set up to transfer momentum along the direction
where the trapped condensate is weakly confined. For relative probe momenta of
k/kF = 3.5 − 9.1, the universal behaviour is measured at the interaction strengths
1/kFa = +0.3, 0, − 0.2. At unitarity, the static structure factor depends linearly
on the momentum transfer. On the BEC side the predicted downwards curvature is
visible while on the BCS side the slight upward bending is confirmed.
Interaction dependent contact
The central parameter for the universal relations is the contact parameter, which
quantifies in the high-momentum tail of the momentum distribution and links the
microscopic details of the atoms to the macroscopic properties of the whole system.
The contact parameter measures the degree of correlations depending inter alia on
the interaction strength. Our previous measurements [80] involved a comprehen-
sive study of the static structure factor of the Bose-Einstein condensate (BEC) to
Bardeen-Cooper-Schrieffer (BCS) crossover at low temperatures using Bragg spec-
troscopy. A smooth transition from molecular to atomic Bragg spectra is observed
with a clear signature of pairing at and above unitarity. Pairing is seen to decay as
the density is lowered, highlighting the many body nature of pair correlations. By
applying sum rules and the universal behaviour of S(k), the interaction dependent
static structure factor is translated into the contact. The contact varies monoton-
ically over the BEC-BCS crossover in good agreement with results and theoretical
predictions of other groups.
Temperature dependent contact
At unitarity the temperature dependence of the contact is measured and described
in chapter 7. The contact starts from the zero-temperature value of around 3.3 and
decreases smoothly but is measurably greater than zero up to T ≈ TF which is
much higher than the critical temperature for superfluidity, TC ≈ 0.2TF . In prin-
ciple, the monotonic decay of the contact data confirms the theoretical predictions.
Unfortunately, the data is not precise enough to favour a certain theoretical model
as different theories predict different values of the contact near TC . The temper-
ature is determined from the empirical fits and compared to the NSR theoretical
calculations. Therefore, the measured temperature is model dependent, but in good
agreement with the measured equation of state [173]. The measurements are re-
stricted to the trapped case, for which the contact does not show a pronounced
peak around the critical temperature region. The large momentum transfer used
CONCLUSION AND OUTLOOK 145
here prevents us from observing the pairing gap.
Outlook
Universal relations open the way for new studies in strongly interacting Fermi gases.
The contact as a new parameter allows pairing mechanisms to be investigated.
Amongst all remaining open questions, the most compelling are probably the pairing
gap and the contact at the critical temperature. The experimental setup is currently
being changed to make contributions to these issues. Improvements of the experi-
mental setup should lead to more accurate Bragg spectra. A new imaging system
with a high quantum efficiency camera with kinetics mode and higher resolution
have been constructed. The imaging beam is now along the axial direction of the
cloud to get symmetric images and reconstructions of the equation of state [237].
Below are some of the directions that may be pursued in future studies. Note that
the suggestions are only concerned with the contact and/or Bragg spectroscopy, al-
though the general idea of strongly interacting Fermi gases forms a very rich testbed
in general.
Bragg spectroscopy above TC
In order to investigate the pseudogap pairing in the critical temperature region near
TC , excitation momenta close to the Fermi wave vector are necessary. With momen-
tum resolved Bragg spectroscopy, the momentum dependence of pair correlations
can be studied. Therefore, the experiments using several lower probe momenta can
search for pairing near the Fermi surface at and above TC . For a momentum trans-
fer of k/kF = 3 − 1, some pairing gap signatures might be apparent. For probe
momenta k/kF < 1, the compressibility sum rule might lead to a measurement of
the speed of sound.
Measurement of the homogeneous contact
The theoretical predictions for the temperature dependent contact show different
results around the critical temperature. It is not clear if the local maximum of the
contact around TC is due to a strengthening of fluctuations [84] or a breakdown of
theoretical models [135]. The differences of the calculations are more pronounced
in the homogeneous case. Spatially resolved Bragg spectroscopy might shed light
on this controversy. Bragg scattering allows information to be extracted in the
linear response regime, where structure factors are model-independent quantities
146 CONCLUSION AND OUTLOOK
and represent a measure of instantaneous fluctuations. Since these fluctuations are
more enhanced in the critical region, the static structure factor should reflect these
divergences in the contact values for the critical temperature region. These studies
require Bragg spectra to be measured with a higher accuracy and precision.
Measurement of new universal quantities
It may be possible to extract new universal relations concerning the dynamic struc-
ture factor with Bragg spectroscopy, such as ∆Sσσ′(k, ω, T ) ∼= ∆Wσσ′(k, ω, T )I,where the Wilson factor, Wσσ′(k, ω, T ), captures the few-body physics in a system
with two spin states, σ [135]. Another universal quantity is the contact for three
identical bosons (trimer) [243]. Eventually, Bragg spectroscopy may have the po-
tential to scatter trimers, if they are stable within an experimental cycle, as the
trimer-frequency could be ωR/3.
Measurement of the contact in different dimensions
Meanwhile, a two-dimensional trap is currently being set up and the higher stability
of pairs in this system will allow for highly interesting and frontier experiments on
pair correlations and novel phenomena in lower dimensions, probed eventually with
Bragg spectroscopy. One candidate would be the spectroscopy of p-wave pairs [244].
Another possibility is again the investigation of the contact in two-dimensional sys-
tems [245]. The dimensional confinement stabilises fluctuations but measurements
of the contact around a critical temperature are speculative at this stage. The con-
tact of one-dimensional systems is also of interest [246] but not specifically planned
in our group.
Determination of the contact from the high-frequency tail
Similar to the rf-spectra one can estimate the contact from the dynamic structure
factor since it is proposed that the high-frequency tail of S(k, ω) has a universal
decay proportional to ω7/2. By multiplying the spectra at low temperatures with
this factor, the high-frequency part approaches a constant value above ω ≈ 2ωR.
This offset is then interpreted as the contact. At unitarity this value is about 5%
of the maximum of S(k, ω) and therefore very difficult to detect experimentally.
However, on the BEC side the contact is larger and it is estimated that around
1/kFa = 1 the contact is about 25% of the maximum of S(k, ω).
Appendix A
Offset locking
For imaging at different high magnetic fields it is convenenient to change the fre-
quency via the LabView control program while the imaging laser stays locked to a
reference laser, despite a large frequency offset. Offset-locking stabilises the beat
signal between the imaging and reference laser and allows the imaging frequency to
be changed over a wide range with a voltage controlled oscillator [247]. The posi-
tive slope of the zero crossing provides the error signal for the laser controller1 of a
second ECDL (ECDL 2). A schematic is given in figureA.1.
The reference frequency is provided by the cooling laser that beats with the
ECDL 2. The grating of the ECDL 2 is pre-aligned to 670.977 nm (D2 line) with a
wavemeter or alternatively via fluorescence of a lithium hollow cathode lamp. The
better the grating is aligned to a minimum lasing threshold, the narrower the beat
signal, and external optical feedback must be well suppressed. The beat signal can
be as narrow as 460 kHz (FWHM) and has a typical frequency in the range fbeat
= 700 − 1000 MHz. The laser beams for the beat signal have a total power of 300
µW and are focused on a custom-made amplifying photodiode. After that the signal
passes another electronic amplifier2. In order to observe the beat signal, a directional
coupler3 sends a small amount to a spectrum analyser4 or frequency counter. The
main signal is electronically mixed5 with a voltage controlled oscillator6, where fV CO
= 612 − 1200 MHz. The frequency difference ∆f = |fbeat − fV CO| is amplified to
a certain power level with an amplifier7 and attenuator8 before being split equally
1MOGlabs Pty Ltd, model DLC-2022Minicircuits: ZX60-3018G-S3Minicircuits: ZX05-5-S4Rohde & Schwarz5Minicircuits: ZX30-17-5-S6Minicircuits: ZX95-1200W-S7Minicircuits: ZX60-3018G-S8Minicircuits: ZX60-3018G-S
147
148 OFFSET LOCKING
with a splitter9. Both signals travel along separate cables before being recombined
with a phase detector10 where the delay line of 1.5 m introduces a delay time, τ ,
and phase shift ϕ = 2π∆fτ . For the delay length of 1.5 m, τ ≈ 7.5 ns. A low pass
filter11 removes the sum frequency above 50 MHz. A load of 500 Ω and 1 µF filters
noise at ∼ 2 kHz. The error signal as a function of the beat frequencies is a series of
cosine fringes, with a period that is set by the length of the delay line. The positive
slope of the main zero crossing is used as an error signal which directly enters the
laser controller box.
This locking can be made very stable if the gain is large enough to provide
a minimum error signal amplitude of 250 mV. The beated lasers do not come out
of lock unless the main ECDL unlocks. The only instabilities are seen when the
VCO voltage is driven close to 1 GHz and the amplifying photodiode is bandwidth
limited.
The voltage range is given by LabView, 0 − 10 V, which covers fbeat ≈ 780
− 1100 MHz. These frequencies provide resonant light for absorption imaging over
the magnetic field range of 700 − 1000 G in the lowest spin state and 650 − 950 G
in the second lowest spin state, respectively. The frequency can be shifted further
with AOMs to be resonant with magnetic fields up to 1200 G.
Since the second ECDL 2 only provides 12 mW, injection seeding of another
diode laser (imaging slave) is necessary. From there, light for the Bragg beams is
provided as well as for both the top and side imaging. Originally, the possibility for
zero-field imaging was included in the design by injecting the slave light into the
imaging slave, see figure 4.4.
9Minicircuits: ZFRSC-205010Minicircuits: ZRPD-111Minicircuits: PLP-50
OFFSET LOCKING 149
Figure A.1: Schematic of the components in the offset locking circuit.
150 OFFSET LOCKING
Bibliography
[1] D. Pines, The Many-Body Problem.
Benjamin, New York, 1962.
[2] H. Onnes, “On the sudden rate at which the resistance of mercury disappears,”
Akad. van Wetenschappen, 1911.
[3] P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas,” Phys. Rev.,
vol. 136, p. 864, 1964.
[4] D. Pines, Elementary excitations in solids.
Benjamin, New York, 1963.
[5] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of Superconductivity,”
Phys. Rev., vol. 108, pp. 1175–1204, 1957.
[6] H. A. Bethe, “Nuclear Many-Body Problem,” Phys. Rev., vol. 103, pp. 1353–
1390, 1956.
[7] K. A. Brueckner, “Two-Body Forces and Nuclear Saturation. III. Details of
the Structure of the Nucleus,” Phys. Rev., vol. 97, pp. 1353–1366, 1955.
[8] D. Pines and M. A. Alpar, “Superfluidity in neutron stars,” Nature, vol. 316,
p. 27, 1985.
[9] M. Anderson, J. Ensher, M. Matthews, E. Wieman, and E. Cornell, “Obser-
vation of Bose-Einstein condensation in a dilute atomic vapor,” Science,
vol. 269, p. 198, 1995.
[10] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle, “Bose-Einstein Condensation in a Gas of
Sodium Atoms,” Phys. Rev. Lett., vol. 75, pp. 3969–3973, 1995.
[11] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose-
Einstein Condensation in an Atomic Gas with Attractive Interactions,”
Phys. Rev. Lett., vol. 75, pp. 1687–1690, 1995.
151
152 BIBLIOGRAPHY
[12] L. Landau, “The Theory of a fermi Liquid,” J. Exptl. theoret. Phys. (U.S.S.R.
), vol. 30, p. 1058, 1956.
[13] L. Landau, “Oscillations in a Fermi Liquid,” J. Exptl. theoret. Phys. (U.S.S.R.
), vol. 32, p. 59, 1957.
[14] L. Landau, “On the theory of the Fermi Liquid,” J. Exptl. theoret. Phys.
(U.S.S.R. ), vol. 35, p. 97, 1958.
[15] V. Galitskii, “The energy spectrum of a non-ideal Fermi Gas,” J. Exptl. the-
oret. Phys. (U.S.S.R. ), vol. 34, p. 151, 1958.
[16] J. Goldstone and K. Gottfried, Collective Excitations of Fermi Gases, ch. In-
teracting Ferion Systems, p. 288.
Benjamin, New York, 1959.
[17] B. DeMarco and D. S. Jin, “Onset of Fermi Degeneracy in a Trapped Atomic
Gas,” Science, vol. 285, p. 1703, 1999.
[18] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. H. Denschlag,
and R. Grimm, “Pure Gas of Optically Trapped Molecules Created from
Fermionic Atoms,” Phys. Rev. Lett., vol. 91, p. 240402, 2003.
[19] F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubi-
zolles, and C. Salomon, “Quasipure Bose-Einstein Condensate Immersed
in a Fermi Sea,” Phys. Rev. Lett., vol. 87, p. 080403, 2001.
[20] A. Truscott, K. S. W. McAlexander, and G. P. R. Huletdagger, “Observation
of Fermi Pressure in a Gas of Trapped Atoms,” Science, vol. 291, p. 2570,
2001.
[21] G. Roati, F. Riboli, G. Modugno, and M. Inguscio, “Fermi-Bose Quantum
Degenerate 40K −87 Rb Mixture with Attractive Interaction,” Phys. Rev.
Lett., vol. 89, p. 150403, 2002.
[22] Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M. W. Zwierlein,
K. Dieckmann, and W. Ketterle, “Fiftyfold Improvement in the Num-
ber of Quantum Degenerate Fermionic Atoms,” Phys. Rev. Lett., vol. 91,
p. 160401, 2003.
[23] D. D. Osheroff, R. C. Richardson, and D. M. Lee, “Evidence for a New Phase
of Solid He3,” Phys. Rev. Lett., vol. 28, pp. 885–888, 1972.
[24] P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J.
Verhaar, “Observation of a Feshbach Resonance in Cold Atom Scattering,”
Phys. Rev. Lett., vol. 81, pp. 69–72, 1998.
BIBLIOGRAPHY 153
[25] S. Inouye, M. Andrews, J. Stenger, H.-J. Miesner, D. Stamper-Kurn, and
W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein con-
densate,” Nature, vol. 392, p. 151, 1998.
[26] J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Pritchard,
and W. Ketterle, “Bragg Spectroscopy of a Bose-Einstein Condensate,”
Phys. Rev. Lett., vol. 82, pp. 4569–4573, 1999.
[27] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, “Weakly Bound Dimers of
Fermionic Atoms,” Phys. Rev. Lett., vol. 93, p. 090404, 2004.
[28] J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and
C. Salomon, “Production of Long-Lived Ultracold Li2 Molecules from a
Fermi Gas,” Phys. Rev. Lett., vol. 91, p. 240401, 2003.
[29] K. E. Strecker, G. B. Partridge, and R. G. Hulet, “Conversion of an Atomic
Fermi Gas to a Long-Lived Molecular Bose Gas,” Phys. Rev. Lett., vol. 91,
p. 080406, 2003.
[30] C. Regal, C. Ticknor, J. Bohn, and D. Jin, “Creation of ultracold molecules
from a Fermi gas of atoms,” Nature, vol. 424, p. 47, 2003.
[31] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Ker-
man, and W. Ketterle, “Condensation of Pairs of Fermionic Atoms near a
Feshbach Resonance,” Phys. Rev. Lett., vol. 92, p. 120403, 2004.
[32] M. Greiner, C. Regal, and D. Jin, “Emergence of a molecular Bose-Einstein
condensate from a Fermi gas,” Nature, vol. 426, p. 537, 2003.
[33] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. H.
Denschlag, and R. Grimm, “Bose-Einstein Condensation of Molecules,”
Science, vol. 302, p. 2101, 2003.
[34] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta,
Z. Hadzibabic, and W. Ketterle, “Observation of Bose-Einstein Condensa-
tion of Molecules,” Phys. Rev. Lett., vol. 91, p. 250401, 2003.
[35] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann,
L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, “Experimental
Study of the BEC-BCS Crossover Region in Lithium 6,” Phys. Rev. Lett.,
vol. 93, p. 050401, 2004.
[36] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, and R. G. Hulet,
“Molecular Probe of Pairing in the BEC-BCS Crossover,” Phys. Rev. Lett.,
vol. 95, p. 020404, 2005.
154 BIBLIOGRAPHY
[37] K. OHara, S. Hemmer, M. Gehm, S. Granade, and J. E. Thomas, “Observa-
tion of a Strongly Interacting Degenerate Fermi Gas of Atoms,” Science,
vol. 298, p. 2179, 2002.
[38] T. Bourdel, J. Cubizolles, L. Khaykovich, K. Magalhaes, S. Kokkelmans,
G. Shlyapnikov, and C. Salomon, “Measurement of the Interaction En-
ergy near a Feshbach Resonance in a 6Li Fermi Gas,” Phys. Rev. Lett.,
vol. 91, p. 020402, 2003.
[39] M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen, S. Jochim, C. Chin,
J. H. Denschlag, R. Grimm, A. Simoni, E. Tiesinga, C. J. Williams, and
P. S. Julienne, “Precise Determination of 6Li Cold Collision Parameters by
Radio-Frequency Spectroscopy on Weakly Bound Molecules,” Phys. Rev.
Lett., vol. 94, p. 103201, 2005.
[40] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag,
and R. Grimm, “Collective Excitations of a Degenerate Gas at the BEC-
BCS Crossover,” Phys. Rev. Lett., vol. 92, p. 203201, 2004.
[41] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. H. Denschlag,
and R. Grimm, “Observation of the Pairing Gap in a Strongly Interacting
Fermi Gas,” Science, vol. 305, p. 1128, 2004.
[42] M. Zwierlein, J. Abo-Shaeer, A. Schirotzek, C. Schunck, and W. Ketterle,
“Vortices and superfluidity in a strongly interacting Fermi gas,” Nature,
vol. 435, p. 1047, 2005.
[43] C. A. Regal, M. Greiner, and D. S. Jin, “Observation of Resonance Condensa-
tion of Fermionic Atom Pairs,” Phys. Rev. Lett., vol. 92, p. 040403, 2004.
[44] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag,
and R. Grimm, “Crossover from a Molecular Bose-Einstein Condensate to
a Degenerate Fermi Gas,” Phys. Rev. Lett., vol. 92, p. 120401, 2004.
[45] J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E. Thomas,
“Evidence for Superfluidity in a Resonantly Interacting Fermi Gas,” Phys.
Rev. Lett., vol. 92, p. 150402, 2004.
[46] A. Altmeyer, S. Riedl, C. Kohstall, M. J. Wright, R. Geursen, M. Bartenstein,
C. Chin, J. H. Denschlag, and R. Grimm, “Precision Measurements of Col-
lective Oscillations in the BEC-BCS Crossover,” Phys. Rev. Lett., vol. 98,
p. 040401, 2007.
BIBLIOGRAPHY 155
[47] J. Kinast, A. Turlapov, and J. E. Thomas, “Breakdown of hydrodynamics in
the radial breathing mode of a strongly interacting Fermi gas,” Phys. Rev.
A, vol. 70, p. 051401, 2004.
[48] J. Kinast, A. Turlapov, J. Thomas, Q.Chen, J.Stajic, and K. Levin, “Heat
Capacity of a Strongly Interacting Fermi Gas,” Science, vol. 307, p. 1296,
2005.
[49] L. Thomas, J. Kinast, and A. Turlapov, “Virial Theorem and Universality in
a Unitary Fermi Gas,” Phys. Rev. Lett., vol. 95, p. 120402, 2005.
[50] L. Luo, B. Clancy, J. Joseph, J. Kinast, and J. Thomas, “Measurement of the
Entropy and Critical Temperature of a Strongly Interacting Fermi Gas,”
Phys. Rev. Lett., vol. 98, p. 080402, 2007.
[51] J. T. Stewart, J. P. Gaebler, C. A. Regal, and D. S. Jin, “Potential Energy of
a 40K Fermi Gas in the BCS-BEC Crossover,” Phys. Rev. Lett., vol. 97,
p. 220406, 2006.
[52] G. Partridge, W.Li, R. Kamar, Y. Liao, and R. Hulet, “Pairing and Phase
Separation in a Polarized Fermi Gas,” Science, vol. 311, p. 503, 2006.
[53] J. K. Chin, D. E. Miller, Y. Liu, C. Stan, W. Setiawan, C. Sanner, K. Xu,
and W. Ketterle, “Evidence for superfluidity of ultracold fermions in an
optical lattice,” Nature, vol. 443, p. 961, 2006.
[54] M. Zwierlein, C. Schunck, A. Schirotzek, and W. Ketterle, “Direct observa-
tion of the superfluid phase transition in ultracold Fermi gases,” Nature,
vol. 442, p. 54, 2006.
[55] C. H. Schunck, M. W. Zwierlein, A. Schirotzek, and W. Ketterle, “Superfluid
Expansion of a Rotating Fermi Gas,” Phys. Rev. Lett., vol. 98, p. 050404,
2007.
[56] C. Schunck, Y. Shin, A. Schirotzek, and W. Ketterle, “Determination of the
fermion pair size in a resonantly interacting superfluid,” Nature, vol. 454,
p. 739, 2008.
[57] M. W. Zwierlein, A. Schirotzek, C. Schunck, and W. Ketterle, “Fermionic
Superfluidity with Imbalanced Spin Populations,” Science, vol. 311, p. 492,
2006.
[58] Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle,
“Observation of Phase Separation in a Strongly Interacting Imbalanced
Fermi Gas,” Phys. Rev. Lett., vol. 97, p. 030401, 2006.
156 BIBLIOGRAPHY
[59] G. B. Partridge, W. Li, Y. A. Liao, R. G. Hulet, M. Haque, and H. T. C. Stoof,
“Deformation of a Trapped Fermi Gas with Unequal Spin Populations,”
Phys. Rev. Lett., vol. 97, p. 190407, 2006.
[60] C. Schunck, Y. Shin, A. Schirotzek, M. W. Zwierlein, and W. Ketterle, “Pair-
ing Without Superfluidity: The Ground State of an Imbalanced Fermi
Mixture,” Science, vol. 316, p. 867, 2007.
[61] Y. Shin, C. Schunck, A. Schirotzek, and W. Ketterle, “Phase diagram of a
two-component Fermi gas with resonant interactions,” Nature, vol. 451,
p. 689, 2008.
[62] J. Joseph, B. Clancy, L. Luo, J. Kinast, A. Turlapov, and J. E. Thomas, “Mea-
surement of Sound Velocity in a Fermi Gas near a Feshbach Resonance,”
Phys. Rev. Lett., vol. 98, p. 170401, 2007.
[63] T.-L. Ho, “Universal Thermodynamics of Degenerate Quantum Gases in the
Unitarity Limit,” Phys. Rev. Lett., vol. 92, p. 090402, 2004.
[64] S. Tan, “Energetics of a strongly correlated Fermi gas,” Ann. Phys., vol. 323,
p. 2952, 2008.
[65] S. Tan, “Large momentum part of a strongly correlated Fermi gas,” Ann.
Phys., vol. 323, p. 2971, 2008.
[66] S. Tan, “Generalised Virial Theorem and Pressure Relation for a strongly
correlated Fermi gas,” Ann. Phys., vol. 323, p. 2987, 2008.
[67] E. Braaten and L. Platter, “Exact Relations for a Strongly Interacting Fermi
Gas from the Operator Product Expansion,” Phys. Rev. Lett., vol. 100,
p. 205301, 2008.
[68] E. Braaten, D. Kang, and L. Platter, “Universal relations for a strongly in-
teracting Fermi gas near a Feshbach resonance,” Phys. Rev. A, vol. 78,
p. 053606, 2008.
[69] F. Werner, “Virial theorems for trapped cold atoms,” Phys. Rev. A, vol. 78,
p. 025601, 2008.
[70] F. Werner, L. Tarruell, and Y. Castin, “Number of closed-channel molecules
in the BEC-BCS crossover,” Eur. Phys. J. B, vol. 68, p. 401, 2009.
[71] F. Werner and Y. Castin, “Exact relations for quantum-mechanical few-body
and many-body problems with short-range interactions in two and three
dimensions,” arXiv:1001.0774v1, 2010.
BIBLIOGRAPHY 157
[72] S. Zhang and A. J. Leggett, “Universal properties of the ultracold Fermi gas,”
Phys. Rev. A, vol. 79, p. 023601, 2009.
[73] R. Combescot, F. Alzetto, and X. Leyronas, “Particle distribution tail and
related energy formula,” Phys. Rev. A, vol. 79, p. 053640, 2009.
[74] S. Tan, “S-wave contact interaction problem: A simple description,”
arXiv:cond-mat/0505615v1, 2010.
[75] R. Combescot, S. Giorgini, and S.Stringari, “Molecular signatures in the struc-
ture factor of an interacing Fermi gas,” Euro. Phys. Lett., vol. 75, p. 695,
2006.
[76] D. Pines and P. Nozires, The Theory of Quantum Liquids, Vol. I.
Benjamin, New York, 1966.
[77] F. Zambelli, L. Pitaevskii, D. M. Stamper-Kurn, and S. Stringari, “Dynamic
structure factor and momentum distribution of a trapped Bose gas,” Phys.
Rev. A, vol. 61, p. 063608, 2000.
[78] D. Stamper-Kurn, A. Chikkatur, a. Goerlitz, S. Inouye, S. Gupta,
D. Pritchard, and W. Ketterle, “Excitations of Phonons in a Bose-Einstein
Condensate by Light Scattering,” Phys. Rev. Lett., vol. 83, p. 2876, 1999.
[79] J. Steinhauer, R. Ozeri, N. Katz, and N. Davidson, “Excitation Spectrum of
a Bose-Einstein Condensate,” Phys. Rev. Lett., vol. 88, p. 120407, 2002.
[80] G. Veeravalli, E. Kuhnle, P. Dyke, and C. J. Vale, “Bragg Spectroscopy of
a Strongly Interacting Fermi Gas,” Phys. Rev. Lett., vol. 101, p. 250403,
2008.
[81] P. W. Anderson, “Random-Phase Approximation in the Theory of Supercon-
ductivity,” Phys. Rev., vol. 112, pp. 1900–1916, 1958.
[82] H. Hu, X.-J. Liu, and P. Drummond, “Static structure factor of a strongly cor-
related Fermi gas at large momenta,” Euro. Phys. Lett., vol. 91, p. 20005,
2010.
[83] H. Hu, X.-J. Liu, and P. Drummond, “Equation of state of a superfluid Fermi
gas in the BCS-BEC crossover,” Euro. Phys. Lett., vol. 74, p. 574, 2006.
[84] F. Palestini, A. Perali, P. Pieri, and G. C. Strinati, “Temperature and coupling
dependence of the universal contact intensity for an ultracold Fermi gas,”
Phys. Rev. A, vol. 82, p. 021605, 2010.
[85] T. Enss, R. Haussmann, and W. Zwerger, “Viscosity and scale invariance in
the unitary Fermi gas,” arXiv:1008.0007v2, 2010.
158 BIBLIOGRAPHY
[86] J. T. Stewart, J. P. Gaebler, and D. S. Jin, “Using photoemission spectroscopy
to probe a strongly interacting Fermi gas,” Nature, vol. 454, p. 744, 2008.
[87] H. T. C. Stoof, M. Houbiers, C. A. Sackett, and R. G. Hulet, “Superfluidity
of Spin-Polarized 6Li,” Phys. Rev. Lett., vol. 76, pp. 10–13, 1996.
[88] E. R. I. Abraham, W. I. McAlexander, J. M. Gerton, R. G. Hulet, R. Cote,
and A. Dalgarno, “Triplet s-wave resonance in 6Li collisions and scattering
lengths of 6Li and 7Li,” Phys. Rev. A, vol. 55, pp. R3299–R3302, 1997.
[89] J. Taylor, Scattering Theory.
John Wiley & Sons, Inc., 1972.
[90] K. Huang and C. N. Yang, “Quantum-Mechanical Many-Body Problem with
Hard-Sphere Interaction,” Phys. Rev., vol. 105, pp. 767–775, 1957.
[91] H. Bethe and R. Peierls, “Quantum Theory of the Diplon,” Proc. R. Soc.
Lond. A, vol. 148, p. 146, 1935.
[92] S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic
Fermi gases,” Rev. Mod. Phys., vol. 80, pp. 1215–1274, 2008.
[93] H. Feshbach, “A Unified Theory of Nuclear Reactions I,” Annals of Physics,
vol. 5, p. 357, 1958.
[94] H. Feshbach, “A Unified Theory of Nuclear Reactions II,” Annals of Physics,
vol. 19, p. 287, 1962.
[95] U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,”
Phys. Rev., vol. 124, pp. 1866–1878, 1961.
[96] W. C. Stwalley, “Stability of Spin-Aligned Hydrogen at Low Temperatures
and High Magnetic Fields: New Field-Dependent Scattering Resonances
and Predissociations,” Phys. Rev. Lett., vol. 37, pp. 1628–1631, 1976.
[97] E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, “Threshold and resonance
phenomena in ultracold ground-state collisions,” Phys. Rev. A, vol. 47,
pp. 4114–4122, 1993.
[98] J. L. Roberts, N. R. Claussen, J. P. Burke, C. H. Greene, E. A. Cornell, and
C. E. Wieman, “Resonant Magnetic Field Control of Elastic Scattering in
Cold 85Rb,” Phys. Rev. Lett., vol. 81, pp. 5109–5112, 1998.
[99] F. H. Mies, E. Tiesinga, and P. S. Julienne, “Manipulation of Feshbach reso-
nances in ultracold atomic collisions using time-dependent magnetic fields,”
Phys. Rev. A, vol. 61, p. 022721, 2000.
BIBLIOGRAPHY 159
[100] F. A. van Abeelen and B. J. Verhaar, “Time-Dependent Feshbach Resonance
Scattering and Anomalous Decay of a Na Bose-Einstein Condensate,”
Phys. Rev. Lett., vol. 83, pp. 1550–1553, 1999.
[101] V. A. Yurovsky, A. Ben-Reuven, P. S. Julienne, and C. J. Williams, “Atom loss
and the formation of a molecular Bose-Einstein condensate by Feshbach
resonance,” Phys. Rev. A, vol. 62, p. 043605, 2000.
[102] E. Donley, N. Claussen, S. Thompson, and C. Wieman, “Atom–molecule co-
herence in a Bose–Einstein condensate,” Nature, vol. 417, p. 529, 2002.
[103] J. Herbig, T. Kraemer, M. Mark, T. Weber, C. Chin, H.-C. Nagerl, and
R. Grimm, “Preparation of a Pure Molecular Quantum Gas,” Science,
vol. 301, p. 1510, 2003.
[104] K. Xu, T. Mukaiyama, J. R. Abo-Shaeer, J. K. Chin, D. E. Miller, and W. Ket-
terle, “Formation of Quantum-Degenerate Sodium Molecules,” Phys. Rev.
Lett., vol. 91, p. 210402, 2003.
[105] S. Duerr, T. Volz, A. Marte, and G. Rempe, “Observation of Molecules
Produced from a Bose-Einstein Condensate,” Phys. Rev. Lett., vol. 92,
p. 020406, 2004.
[106] K. Dieckmann, C. A. Stan, S. Gupta, Z. Hadzibabic, C. H. Schunck, and
W. Ketterle, “Decay of an Ultracold Fermionic Lithium Gas near a Fesh-
bach Resonance,” Phys. Rev. Lett., vol. 89, p. 203201, 2002.
[107] E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S.
Jin, E. A. Cornell, and C. E. Wieman, “Production Efficiency of Ultracold
Feshbach Molecules in Bosonic and Fermionic Systems,” Phys. Rev. Lett.,
vol. 94, p. 120402, 2005.
[108] A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and
P. Pillet, “Formation of Cold Cs2 Molecules through Photoassociation,”
Phys. Rev. Lett., vol. 80, pp. 4402–4405, 1998.
[109] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in
ultracold gases,” Rev. Mod. Phys., vol. 82, pp. 1225–1286, 2010.
[110] M. Houbiers, H. T. C. Stoof, W. I. McAlexander, and R. G. Hulet, “Elastic
and inelastic collisions of 6Li atoms in magnetic and optical traps,” Phys.
Rev. A, vol. 57, pp. R1497–R1500, 1998.
[111] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, “Scattering properties of
weakly bound dimers of fermionic atoms,” Phys. Rev. A, vol. 71, p. 012708,
2005.
160 BIBLIOGRAPHY
[112] W. Ketterle and M. Zwierlein, “Making, probing and understanding ultracold
Fermi gases,” in Ultra-cold Fermi Gases, Proceedings of the international
school of physics ”Enrico Fermi”, pp. 95–274, Societa Italiana Di Fisica
Bologna-Italy, 2007.
[113] C. Menotti, P. Pedri, and S. Stringari, “Expansion of an Interacting Fermi
Gas,” Phys. Rev. Lett., vol. 89, p. 250402, 2002.
[114] C. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases.
Cambridge University Press, 2001.
[115] H. Heiselberg, “Fermi systems with long scattering lengths,” Phys. Rev. A,
vol. 63, p. 043606, 2001.
[116] T.-L. Ho and E. J. Mueller, “High Temperature Expansion Applied to
Fermions near Feshbach Resonance,” Phys. Rev. Lett., vol. 92, p. 160404,
2004.
[117] H. Hu, P. Drummond, and J.-X. Liu, “Universal thermodynamics of strongly
interacting Fermi gases,” Nat. Phys., vol. 3, p. 469, 2007.
[118] G. A. Baker, “Neutron matter model,” Phys. Rev. C, vol. 60, p. 054311, 1999.
[119] A. Gezerlis and J. Carlson, “Strongly paired fermions: Cold atoms and neutron
matter,” Phys. Rev. C, vol. 77, p. 032801, 2008.
[120] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics.
John Wiley & Sons, Inc, 1960.
[121] O. Klein, “Mesons and nucleons,” Nature, vol. 161, p. 897, 1948.
[122] G. Puppi, “Sui mesoni dei raggi cosmici,” Nuovo cimento, vol. 5, p. 587, 1948.
[123] T. D. Lee, M. Rosenbluth, and C. N. Yang, “Interaction of Mesons with Nu-
cleons and Light Particles,” Phys. Rev., vol. 75, p. 905, 1949.
[124] J. Tiomno and J. A. Wheeler, “Energy Spectrum of Electrons from Meson
Decay,” Rev. Mod. Phys., vol. 21, pp. 144–152, 1949.
[125] E. Braaten, “Universal Relations for Fermions with Large Scattering Length,”
arXiv:1008.2922v1, 2011.
[126] L. Landau and E. Lifshitz, Statistical Physics, Third edition Part 1.
Pergamon Press, 1980.
[127] L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Sov. Phys. JETP,
vol. 13, p. 451, 1961.
BIBLIOGRAPHY 161
[128] E. P. Gross, “Structure of a quantized vortex in boson systems,” Nuovo Ci-
mento, vol. 20, p. 454, 1961.
[129] E. Gross, “Hydrodynamics of a superfluid condensate,” Journal of Mathemat-
ical Physics, vol. 4, p. 195, 1963.
[130] N. Navon, S. Nascimbne, F. Chevy, and C. Salomon, “The Equation of State
of a Low-Temperature Fermi Gas with Tunable Interactions,” Science,
vol. 328, p. 729, 2010.
[131] S. Gandolfi, K. Schmidt, and J. Carlson, “BEC-BCS crossover and universal
relations in unitary Fermi gases,” arXiv:1012.4417v1, 2010.
[132] J. Drut, T. Lahde, and T. Ten, “Momentum Distribution and Contact of the
Unitay Fermi gas,” arXiv:1012.5474v1, 2010.
[133] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, “Verification of
Universal Relations in a Strongly Interacting Fermi Gas,” Phys. Rev. Lett.,
vol. 104, p. 235301, 2010.
[134] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-
Einstein condensation in trapped gases,” Rev. Mod. Phys., vol. 71, pp. 463–
512, 1999.
[135] H. Hu, X.-J. Liu, and P. Drummond, “Universal contact of strongly interacting
fermions at finite temperatures,” arXiv:1011.3845v1, 2010.
[136] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Han-
naford, and C. J. Vale, “Universal Behavior of Pair Correlations in a
Strongly Interacting Fermi Gas,” Phys. Rev. Lett., vol. 105, p. 070402,
2010.
[137] T. D. Lee and C. N. Yang, “Many-Body Problem in Quantum Mechanics
and Quantum Statistical Mechanics,” Phys. Rev., vol. 105, pp. 1119–1120,
1957.
[138] T. D. Lee, K. Huang, and C. N. Yang, “Eigenvalues and Eigenfunctions of a
Bose System of Hard Spheres and its Low-Temperature Properties,” Phys.
Rev., vol. 106, pp. 1135–1145, 1957.
[139] X. Leyronas and R. Combescot, “Superfluid Equation of State of Dilute Com-
posite Bosons,” Phys. Rev. Lett., vol. 99, p. 170402, 2007.
[140] E. Braaten, H.-W. Hammer, and T. Mehen, “Dilute Bose-Einstein Condensate
with Large Scattering Length,” Phys. Rev. Lett., vol. 88, p. 040401, 2002.
162 BIBLIOGRAPHY
[141] A. Bulgac and G. F. Bertsch, “Collective Oscillations of a Trapped Fermi Gas
near the Unitary Limit,” Phys. Rev. Lett., vol. 94, p. 070401, 2005.
[142] S. Y. Chang, V. R. Pandharipande, J. Carlson, and K. E. Schmidt, “Quantum
Monte Carlo studies of superfluid Fermi gases,” Phys. Rev. A, vol. 70,
p. 043602, 2004.
[143] G. E. Astrakharchik, J. Boronat, J. Casulleras, Giorgini, and S. Giorgini,
“Equation of State of a Fermi Gas in the BEC-BCS Crossover: A Quantum
Monte Carlo Study,” Phys. Rev. Lett., vol. 93, p. 200404, 2004.
[144] G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, “Momentum
Distribution and Condensate Fraction of a Fermion Gas in the BCS-BEC
Crossover,” Phys. Rev. Lett., vol. 95, p. 230405, 2005.
[145] C. Lobo, I. Carusotto, S. Giorgini, A. Recati, and S. Stringari, “Pair Corre-
lations of an Expanding Superfluid Fermi Gas,” Phys. Rev. Lett., vol. 97,
p. 100405, 2006.
[146] R. B. Diener, R. Sensarma, and M. Randeria, “Quantum fluctuations in the su-
perfluid state of the BCS-BEC crossover,” Phys. Rev. A, vol. 77, p. 023626,
2008.
[147] G. A. Baker, “Singularity Structure of the Perturbation Series for the Ground-
State Energy of a Many-Fermion System,” Rev. Mod. Phys., vol. 43,
pp. 479–531, 1971.
[148] Z. Yu and G. Baym, “Spin-correlation functions in ultracold paired atomic-
fermion systems: Sum rules, self-consistent approximations, and mean
fields,” Phys. Rev. A, vol. 73, p. 063601, 2006.
[149] H. Hu, X.-J. Liu, and P. D. Drummond, “Dynamic response of strongly corre-
lated Fermi gases in the quantum virial expansion,” Phys. Rev. A, vol. 81,
p. 033630, 2010.
[150] J. Steinhauer, N. Katz, R. Ozeri, N. Davidson, C. Tozzo, and F. Dalfovo,
“Bragg Spectroscopy of a Multibranch Bogoliubov Spectrum of Elongated
Bose-Einstein Condensates,” Phys. Rev. Lett., vol. 90, p. 060404, 2003.
[151] S. Zhang and A. J. Leggett, “Sum-rule analysis of radio-frequency spectroscopy
of ultracold Fermi gas,” Phys. Rev. A, vol. 77, p. 033614, 2008.
[152] J. Carlson and S. Reddy, “Superfluid Pairing gap in Strong Coupling,” Phys.
Rev. Lett., vol. 100, p. 150403, 2008.
BIBLIOGRAPHY 163
[153] A. Schirotzek, Y. Shin, C. Schunck, and W. Ketterle, “Determination of the
Superfluid Gap in Atomic Fermi Gases by Quasiparticle Spectroscopy,”
Phys. Rev. Lett., vol. 101, p. 140403, 2008.
[154] V. M. Loktev, R. M. Quick, and S. G. Sharapov, “Phase fluctuations and
pseudogap phenomena,” Physics Reports, vol. 349, pp. 1 – 123, 2001.
[155] Y. Yanase, T. Jujo, T. Nomura, H. Ikeda, T. Hotta, and K. Yamada, “The-
ory of superconductivity in strongly correlated electron systems,” Physics
Reports, vol. 387, pp. 1 – 149, 2003.
[156] R. Haussmann, “Properties of a Fermi liquid at the superfluid transition in
the crossover region between BCS superconductivity and Bose-Einstein
condensation,” Phys. Rev. B, vol. 49, pp. 12975–12983, 1994.
[157] X.-J. Liu and H. Hu, “Self-consistent theory of atomic Fermi gases with a
feshbach resonance at the superfluid transition,” Phys. Rev. A, vol. 72,
p. 063613, 2005.
[158] R. Haussmann, W. Rantner, S. Cerrito, and W. Zwerger, “Thermodynamics
of the BCS-BEC crossover,” Phys. Rev. A, vol. 75, p. 023610, 2007.
[159] A. Perali, P. Pieri, G. C. Strinati, and C. Castellani, “Pseudogap and spectral
function from superconducting fluctuations to the bosonic limit,” Phys.
Rev. B, vol. 66, p. 024510, 2002.
[160] P. Pieri, L. Pisani, and G. C. Strinati, “BCS-BEC crossover at finite temper-
ature in the broken-symmetry phase,” Phys. Rev. B, vol. 70, p. 094508,
2004.
[161] R. Combescot, X. Leyronas, and M. Y. Kagan, “Self-consistent theory for
molecular instabilities in a normal degenerate Fermi gas in the BEC-BCS
crossover,” Phys. Rev. A, vol. 73, p. 023618, 2006.
[162] P. Nozieres and S. Schmitt-Rink, “Bose condensation in an attractive fermion
gas: From weak to strong coupling superconductivity,” Journal of Low
Temperature Physics, vol. 59, pp. 195–211, 1985.
[163] S. de Melo, C. A. R., Randeria, M., and J. Engelbrecht, “Crossover from BCS
to Bose superconductivity: Transition temperature and time-dependent
Ginzburg-Landau theory,” Phys. Rev. Lett., vol. 71, pp. 3202–3205, 1993.
[164] X.-J. Liu and H. Hu, “BCS-BEC crossover in an asymmetric two-component
Fermi gas,” Euro. Phys. Lett., vol. 75, p. 364, 2006.
164 BIBLIOGRAPHY
[165] Z. Yu, G. M. Bruun, and G. Baym, “Short-range correlations and entropy in
ultracold-atom Fermi gases,” Phys. Rev. A, vol. 80, p. 023615, 2009.
[166] H. Hu, X.-J. Liu, and P. D. Drummond, “Comparative study of strong-
coupling theories of a trapped Fermi gas at unitarity,” Phys. Rev. A,
vol. 77, p. 061605, 2008.
[167] H. Hu, X.-J. Liu, and P. Drummond, “Universal thermodynamics of a strongly
interacting Fermi gas: theory versus experiment,” New J. Phys., vol. 12,
p. 063038, 2010.
[168] R. Haussmann, M. Punk, and W. Zwerger, “Spectral functions and rf response
of ultracold fermionic atoms,” Phys. Rev. A, vol. 80, p. 063612, 2009.
[169] X.-J. Liu, H. Hu, and P. D. Drummond, “Virial Expansion for a Strongly
Correlated Fermi Gas,” Phys. Rev. Lett., vol. 102, p. 160401, 2009.
[170] H. Hu, X.-J. Liu, P. Drummond, and H. Dong, “Pseudogap Pairing in Ultra-
cold Fermi Atoms,” Phys. Rev. Lett., vol. 104, p. 240407, 2010.
[171] X.-J. Liu, H. Hu, and P. D. Drummond, “Three attractively interacting
fermions in a harmonic trap: Exact solution, ferromagnetism, and high-
temperature thermodynamics,” Phys. Rev. A, vol. 82, p. 023619, 2010.
[172] E. Beth and G. Uhlenbeck, “The quantum theory of the non-ideal gas. II.
Behaviour at low temperatures,” Physica, vol. 4, p. 915, 1937.
[173] S. Nascimbne, N. Navon, K. Jiang, F. Chevy, and C. Salomon, “Exploring the
thermodynamics of a universal Fermi gas,” Nature, vol. 463, p. 1057, 2010.
[174] L. Pitaevskii and S. Stringari, Bose-Einstein condensation.
Clarendon Press, 2003.
[175] C. Cohen-Tannoudji and C. Robilliard, “Wave functions, relative phase and
interference for atomic Bose–Einstein condensates,” Comptes Rendus de
l’Acadmie des Sciences - Series IV - Physics, vol. 2, p. 445, 2001.
[176] J. Javanainen, “Spectrum of Light Scattered from a Degenerate Bose Gas,”
Phys. Rev. Lett., vol. 75, pp. 1927–1930, 1995.
[177] R. Graham and D. Walls, “Spectrum of Light Scattered from a Weakly Inter-
acting Bose-Einstein Condensed Gas,” Phys. Rev. Lett., vol. 76, pp. 1774–
1775, 1996.
[178] A. Parkins and D. Walls, “The physics of trapped dilute-gas Bose-Einstein
condensates,” Physics Reports, vol. 303, p. 1, 1998.
BIBLIOGRAPHY 165
[179] H. D. Politzer, “Bose-stimulated scattering off a cold atom trap,” Phys. Rev.
A, vol. 55, pp. 1140–1146, 1997.
[180] P. J. Martin, B. G. Oldaker, A. H. Miklich, and D. E. Pritchard, “Bragg
scattering of atoms from a standing light wave,” Phys. Rev. Lett., vol. 60,
pp. 515–518, 1988.
[181] D. M. Giltner, R. W. McGowan, and S. A. Lee, “Atom Interferometer Based
on Bragg Scattering from Standing Light Waves,” Phys. Rev. Lett., vol. 75,
pp. 2638–2641, 1995.
[182] S. Bernet, M. K. Oberthaler, R. Abfalterer, J. Schmiedmayer, and A. Zeilinger,
“Coherent Frequency Shift of Atomic Matter Waves,” Phys. Rev. Lett.,
vol. 77, pp. 5160–5163, 1996.
[183] M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L.
Rolston, and W. D. Phillips, “Coherent Splitting of Bose-Einstein Con-
densed Atoms with Optically Induced Bragg Diffraction,” Phys. Rev. Lett.,
vol. 82, pp. 871–875, 1999.
[184] D. Greif, L. Tarruell, T. Uehlinger, R. Jordens, and T. Esslinger, “Probing
nearest-neighbor correlations of ultracold fermions in an optical lattice,”
arXiv:1012.0845v1, 2010.
[185] P. Ernst, S. Gotze, J. Krauser, K. Pyka, D. Luhmann, D. Pfannkuche,
and K. Sengstock, “Probing superfluids in optical lattices by momentum-
resolved Bragg spectroscopy,” Nature Physics, vol. 6, p. 56, 2010.
[186] D. Clement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Multi-band
spectroscopy of inhomogeneous Mott-insulator states of ultracold bosons,”
New J. Phys., vol. 11, p. 103030, 2009.
[187] D. Clement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Cor-
related 1D Bose Gases from the Superfluid to the Mott-Insulator State by
Inelastic Light Scattering,” Phys. Rev. Lett., vol. 102, p. 155301, 2009.
[188] P. C. Hohenberg and P. M. Platzman, “High-Energy Neutron Scattering from
Liquid He4,” Phys. Rev., vol. 152, pp. 198–200, 1966.
[189] G. Veeravalli, Bragg spectroscopy of a strongly interacting Fermi gas.
PhD thesis, Swinburne University of Technology, 2009.
[190] H. A. Kramers, “La diffusion de la lumiere par les atomes,” Atti Congr. Fisica
Como, vol. 2, p. 545, 1927.
166 BIBLIOGRAPHY
[191] R. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am, vol. 12,
p. 547, 1926.
[192] P. Zou, E. D. Kuhnle, C. J. Vale, and H. Hu, “Quantitative comparison
between theoretical predictions and experimental results for Bragg spec-
troscopy of a strongly interacting Fermi superfluid,” Phys. Rev. A, vol. 82,
p. 061605, 2010.
[193] H. Nyquist, “Thermal Agitation of Electric Charge in Conductors,” Phys.
Rev., vol. 32, p. 110, 1928.
[194] H. Callen and T. Welton, “Irreversibility and Generalized Noise,” Phys. Rev.,
vol. 83, p. 34, 1951.
[195] R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I. General
Theory and Simple Applications to Magnetic and Conduction Problems,”
Journal of the Physical Society of Japan, vol. 12, pp. 570–586, 1957.
[196] J. M. Pino, R. J. Wild, P. Makotyn, D. S. Jin, and E. A. Cornell, “Photon
counting for bragg spectroscopy of quantum gases,” Phys. Rev. A, vol. 83,
p. 033615, Mar 2011.
[197] A. Brunello, F. Dalfovo, L. Pitaevskii, S. Stringari, and F. Zambelli, “Momen-
tum transferred to a trapped Bose-Einstein condensate by stimulated light
scattering,” Phys. Rev. A, vol. 64, p. 063614, 2001.
[198] J. Fuchs, Molecular Bose-Einstein condensates and p-wave Feshbach molecules
of 6Li2.
PhD thesis, Swinburne University of Technology, 2009.
[199] J. Fuchs, G. J. Duffy, G. Veeravalli, P. Dyke, M. Bartenstein, C. J. Vale,
P. Hannaford, andW. J. Rowlands, “Molecular Bose-Einstein condensation
in a versatile low power crossed dipole trap,” J. Phys. B, vol. 40, p. 4109,
2007.
[200] P. Dyke, Quasi Two-Dimensional 6Li Fermi gas.
PhD thesis, Swinburne University of Technology, 2010.
[201] H. Metcalf and P. Straaten, Laser Cooling and Trapping.
Springer-Verlag, Heidelberg, 1999.
[202] M. Inguscio, W. Ketterle, and C. Salomon, Ultra-cold Fermi Gases, Proceed-
ings of the international school of physics ”Enrico Fermi”.
IOS Press, 2007.
BIBLIOGRAPHY 167
[203] C. Silber, S. Gunther, C. Marzok, B. Deh, P. W. Courteille, and C. Zimmer-
mann, “Quantum-Degenerate Mixture of Fermionic Lithium and Bosonic
Rubidium Gases,” Phys. Rev. Lett., vol. 95, p. 170408, 2005.
[204] S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental Observation
of Optically Trapped Atoms,” Phys. Rev. Lett., vol. 57, pp. 314–317, 1986.
[205] R. Grimm, M. Weidemuller, and Y. B. Ovchinnikov, “Optical Dipole Traps
for Neutral Atoms,” Advances In Atomic, Molecular, and Optical Physics,
vol. 42, p. 95, 2000.
[206] W. Ketterle, D. Durfee, and D. Stamper-Kurn, “Making, probing and under-
standing Bose-Einstein condensates,” in Bose-Einstein Condensation in
Atomic Gases Proceedings of the International School of Physics (Enrico
Fermi), Varenna 1998, p. 67, 1999.
[207] M. DePue, S. L. Winoto, D. J. Han, and D. Weiss, “Transient compression of
a MOT and high intensity fluorescent imaging of optically thick clouds of
atoms,” Optics Communications, vol. 180, p. 73, 2000.
[208] E. Hecht, Optics, 2nd edition.
Addison Wesley Publishing Company, 1990.
[209] M. R. Andrews, D. M. Kurn, H.-J. Miesner, D. S. Durfee, C. G. Townsend,
S. Inouye, and W. Ketterle, “Propagation of Sound in a Bose-Einstein
Condensate,” Phys. Rev. Lett., vol. 79, pp. 553–556, 1997.
[210] V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, “Reconstruc-
tion of Abel-transformable images: The Gaussian basis-set expansion Abel
transform method,” Review of Scientific Instruments, vol. 73, p. 2634,
2002.
[211] R. Scheunemann, F. S. Cataliotti, T. W. Hansch, and M. Weitz, “Resolving
and addressing atoms in individual sites of a CO2-laser optical lattice,”
Phys. Rev. A, vol. 62, p. 051801, 2000.
[212] T. Gericke, P. Wrtz, D. Reitz, T. Langen, and H. Ott, “High-resolution scan-
ning electron microscopy of an ultracold quantum gas,” Nature Physics,
vol. 4, p. 949, 2008.
[213] W. Bakr, J. Gillen, A. Peng, S. Flling, and M. Greiner, “A quantum gas
microscope for detecting single atoms in a Hubbard-regime optical lattice,”
Nature, vol. 462, p. 74, 2009.
168 BIBLIOGRAPHY
[214] J. Sherson, C.Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr,
“Single-atom-resolved fluorescence imaging of an atomic Mott insulator,”
Nature, vol. 467, p. 68, 2010.
[215] G. Reinaudi, T. Lahaye, Z. Wang, and D. Gury-Odelin, “Strong saturation
absorption imaging of dense clouds of ultracold atoms,” Optics Letters,
vol. 32, p. 3143, 2007.
[216] M. Erhard, Experimente mit mehrkomponentigen Bose-Einstein-Kondensaten.
PhD thesis, Universitat Hamburg, 2004.
[217] J. Kronjager, Coherent Dynamics of Spinor Bose-Einstein Condensates.
PhD thesis, Universitat Hamburg, 2007.
[218] A. Griesmaier, Bose-Einstein condensation of chromium atoms.
PhD thesis, Universitat Stuttgart, 2007.
[219] C. F. Ockeloen, A. F. Tauschinsky, R. J. C. Spreeuw, and S. Whitlock, “De-
tection of small atom numbers through image processing,” Phys. Rev. A,
vol. 82, p. 061606, 2010.
[220] J. Kinnunen, M. Rodriguez, and P. Torma, “Pairing Gap and In-Gap Ex-
citations in Trapped Fermionic Superfluids,” Science, vol. 305, p. 1131,
2004.
[221] A. Leggett, “What Do we know about high Tc,” Nat. Phys., vol. 2, p. 134,
2006.
[222] V. B. Geshkenbein, L. B. Ioffe, and A. I. Larkin, “Superconductivity in a
system with preformed pairs,” Phys. Rev. B, vol. 55, pp. 3173–3180, 1997.
[223] J. Stajic, J. N. Milstein, Q. Chen, M. L. Chiofalo, M. J. Holland, and K. Levin,
“Nature of superfluidity in ultracold Fermi gases near Feshbach reso-
nances,” Phys. Rev. A, vol. 69, p. 063610, 2004.
[224] M. Eagles, “Possible Pairing without Superconductivity at Low Carrier Con-
centrations in Bulk and Thin-Film Superconducting Semiconductors,”
Phys. Rev. Lett., vol. 186, p. 456, 1969.
[225] Q. Chen, J. Stajic, and K. Levin, “Thermodynamics of Interacting Fermions
in Atomic Traps,” Phys. Rev. Lett., vol. 95, p. 260405, 2005.
[226] A. Leggett, Modern Trends in the Theory of Condesed Matter.
Springer-Verlag, Berlin, 1980.
BIBLIOGRAPHY 169
[227] J. Carlson, S.-Y. Chang, V. R. Pandharipande, and K. E. Schmidt, “Superfluid
Fermi Gases with Large Scattering Length,” Phys. Rev. Lett., vol. 91,
p. 050401, 2003.
[228] M. E. Gehm, S. L. Hemmer, S. R. Granade, K. M. O’Hara, and J. E. Thomas,
“Mechanical stability of a strongly interacting Fermi gas of atoms,” Phys.
Rev. A, vol. 68, p. 011401, Jul 2003.
[229] A. Bulgac, J. Drut, and P. Magierski, “Spin 1/2 Fermions in the Unitary
Regime: A Superfluid of a New Type,” Phys. Rev. Lett., vol. 96, p. 090404,
2006.
[230] R. Haussmann and W. Zwerger, “Thermodynamics of a trapped unitary Fermi
gas,” Phys. Rev. A, vol. 78, p. 063602, 2008.
[231] L. Luo and J. Thomas, “Thermodynamic Measurements in a Strongly Inter-
acting Fermi Gas,” J. Low Temp. Phys., vol. 154, p. 1, 2009.
[232] Q. Chen, C. A. Regal, D. S. Jin, and K. Levin, “Finite-temperature momentum
distribution of a trapped Fermi gas,” Phys. Rev. A, vol. 74, p. 011601, 2006.
[233] C. A. Regal, M. Greiner, S. Giorgini, M. Holland, and D. S. Jin, “Momentum
Distribution of a Fermi Gas of Atoms in the BCS-BEC Crossover,” Phys.
Rev. Lett., vol. 95, p. 250404, 2005.
[234] L. D. Carr, G. V. Shlyapnikov, and Y. Castin, “Achieving a BCS Transition
in an Atomic Fermi Gas,” Phys. Rev. Lett., vol. 92, p. 150404, 2004.
[235] H. Hu, X.-J. Liu, and P. Drummond, “Temperature of a trapped unitary
Fermi gas at finite entropy,” Phys. Rev. A, vol. 73, p. 023617, 2006.
[236] C. Cao, E. Elliott, J. Joseph, H. Wu, J. Petricka, T. Schfer, and J. E. Thomas,
“Universal Quantum Viscosity in a Unitary Fermi Gas,” Science, vol. 331,
p. 58, 2011.
[237] T.-L. Ho and Q. Zhou, “Obtaining the phase diagram and thermodynamic
quantities of bulk systems from the densities of trapped gases,” Nat. Phys.,
vol. 6, p. 131, 2010.
[238] S. B. Papp, J. M. Pino, R. J. Wild, S. Ronen, C. E. Wieman, D. S. Jin,
and E. A. Cornell, “Bragg Spectroscopy of a Strongly Interacting 85Rb
Bose-Einstein Condensate,” Phys. Rev. Lett., vol. 101, p. 135301, 2008.
[239] A. Perali, P. Pieri, and G. C. Strinati, “Quantitative Comparison be-
tween Theoretical Predictions and Experimental Results for the BCS-BEC
Crossover,” Phys. Rev. Lett., vol. 93, p. 100404, 2004.
170 BIBLIOGRAPHY
[240] J. Gaebler, J. Stewart, T. Drake, D. Jin, A. Perali, P. Pieri, and G. Strinati,
“Observation of pseudogap behaviour in a strongly interacting Fermi gas,”
Nature Physics, vol. 6, p. 569, 2010.
[241] A. Perali, P.Pieri, F. Palestini, G. Strinati, J. Stewart, J. Gaebler, T. Drake,
and D. Jin, “Pseudogap Phase and Remnant Fermi Surface in Ultracold
Fermi Gases,” arXiv:1006:3406v1, 2010.
[242] A. Kanigel, U. Chatterjee, M. Randeria, M. R. Norman, G. Koren, K. Kad-
owaki, and J. C. Campuzano, “Evidence for Pairing above the Transition
Temperature of Cuprate Superconductors from the Electronic Dispersion
in the Pseudogap Phase,” Phys. Rev. Lett., vol. 101, p. 137002, 2008.
[243] E. Braaten, D. Kang, and L. Platter, “Universal Relations for Identical Bosons
from 3-Body Physics,” arXiv:1101.2854v1, 2011.
[244] J. Fuchs, C. Ticknor, P. Dyke, G. Veeravalli, E. Kuhnle, W. Rowlands,
P. Hannaford, and C. J. Vale, “Binding energies of 6Li p-wave Feshbach
molecules,” Phys. Rev. A, vol. 77, p. 053616, 2008.
[245] G. Bertaina and S. Giorgini, “BCS-BEC crossover in a two-dimensional Fermi
gas,” arXiv:1011.3737v1, 2010.
[246] M. Barth and W. Zwerger, “Tan relations in one dimension,”
arXiv:1101.5594v1, 2011.
[247] U. Schunemann, H. Engler, R. Grimm, M. Weidemuller, and M. Zielonkowski,
“Simple scheme for tunable frequency offset locking of two lasers,” Review
of Scientific Instruments, vol. 70, pp. 242 –243, 1999.
List of publications
During my time as a PhD student, I was co-author in the following papers:
Temperature Dependence of the Universal Contact Parameter in a Unitary Fermi
Gas
E.D. Kuhnle, S. Hoinka, P. Dyke, H. Hu and C.J. Vale
Phys. Rev. Lett. 106, 170402 (2011).
Studies of the universal contact in a strongly interacting Fermi gas using Bragg
spectroscopy
E.D. Kuhnle, S. Hoinka, H. Hu, P. Dyke, P. Hannaford and C.J. Vale
New J. Phys. 13, 055010 (2011).
Universal behaviour of Pair Correlations in a Strongly Interacting Fermi Gas
E.D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P.D. Drummond, P. Hannaford
and C.J. Vale
Phys. Rev. Lett. 105, 070402 (2010).
Bragg Spectroscopy of a Strongly Interacting Fermi Gas
G. Veeravalli, E.D. Kuhnle, P. Dyke and C.J. Vale
Phys. Rev. Lett. 101, 250403 (2008).
Quantitative comparison between theoretical predictions and experimental results
for Bragg spectroscopy of a strongly interacting Fermi superfluid
P. Zou, E.D. Kuhnle, H. Hu and C.J. Vale
Phys. Rev. A(R) 82, 061605 (2010).
Universal structure of a strongly interacting Fermi gas
E.D. Kuhnle, S. Hoinka, P. Dyke, H. Hu, P. Hannaford and C.J. Vale
In proc. ICAP XXII (2010).
171
172 LIST OF PUBLICATIONS
Crossover from 2D to 3D in a weakly interacting Fermi gas
P. Dyke, E.D. Kuhnle, S. Whitlock, H. Hu, M. Mark, S. Hoinka, M. Lingham, P.
Hannaford and C.J. Vale
Phys. Rev. Lett. 106, 105304, (2010).
Binding energies of 6Li p-wave Feshbach molecules
J. Fuchs, C. Ticknor, P. Dyke, G. Veeravalli, E.D. Kuhnle, P. Dyke, W. Rowlands,
P. Hannaford and C.J. Vale
Phys. Rev. A 77, 053616 (2008).